Identifying similar triangles in the coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:04.890
Which of the triangles, triangle
DEF, triangle GHI, and triangle

00:00:04.890 --> 00:00:08.250
JKL, are similar
to triangle ABC?

00:00:08.250 --> 00:00:10.030
And we have this triangle ABC.

00:00:10.030 --> 00:00:13.180
They've given us, or they've
just marked one of the angles,

00:00:13.180 --> 00:00:15.990
and then they've given the
lengths of the two sides

00:00:15.990 --> 00:00:18.290
on either side of that angle.

00:00:18.290 --> 00:00:20.500
And you might remember
from geometry class,

00:00:20.500 --> 00:00:22.390
you can prove that
two triangles are

00:00:22.390 --> 00:00:27.310
similar if you can show they
have corresponding congruent

00:00:27.310 --> 00:00:27.810
angles.

00:00:27.810 --> 00:00:29.309
So let's say we're
trying to compare

00:00:29.309 --> 00:00:36.990
this triangle to this triangle,
triangle JKL to triangle ABC.

00:00:36.990 --> 00:00:39.460
So if you can show that they
have two corresponding angles

00:00:39.460 --> 00:00:41.410
that are congruent
to each other,

00:00:41.410 --> 00:00:45.870
and that the sides on
either side, the ratio

00:00:45.870 --> 00:00:50.830
between the sides on either
side of that angle are the same,

00:00:50.830 --> 00:00:54.160
then you're going to be dealing
with a similar triangle.

00:00:54.160 --> 00:00:58.044
So, for example, if we say
that this angle corresponds

00:00:58.044 --> 00:00:59.460
to this angle, and
they've already

00:00:59.460 --> 00:01:01.126
marked it to say that
they're congruent,

00:01:01.126 --> 00:01:03.100
so that's a pretty
good starting point.

00:01:03.100 --> 00:01:08.550
And if we could show that
the ratio between this side

00:01:08.550 --> 00:01:17.450
and this side, side KL is the
same as the ratio between BA

00:01:17.450 --> 00:01:21.900
and KJ, then we can
say that these two

00:01:21.900 --> 00:01:22.910
are similar triangles.

00:01:22.910 --> 00:01:24.410
Now you might be
thinking, well, how

00:01:24.410 --> 00:01:29.930
did I decide to compare
BC to KL and BA to KJ?

00:01:29.930 --> 00:01:34.040
Well, both of these are the two
sides that essentially make up

00:01:34.040 --> 00:01:37.090
this angle, and BC is
the shorter of the two.

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These are the two sides
that make up this angle,

00:01:41.090 --> 00:01:43.110
both of these are the,
I guess you could say,

00:01:43.110 --> 00:01:46.270
they're segments
of the rays, there

00:01:46.270 --> 00:01:49.110
are parts of the rays that
make up this angle if you were

00:01:49.110 --> 00:01:54.840
to continue forever, and the
blue one is the shorter one,

00:01:54.840 --> 00:01:57.570
so that's the best candidate.

00:01:57.570 --> 00:02:02.540
KL is the best candidate
to correspond to BC.

00:02:02.540 --> 00:02:04.150
Well in order to do
this, we're going

00:02:04.150 --> 00:02:06.210
to figure out the
lengths of these sides.

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We're going to have
to do it for all

00:02:07.710 --> 00:02:09.181
of these triangles
right over here.

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And they don't give
us the lengths,

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but this thing is sitting
on a coordinate plane.

00:02:13.730 --> 00:02:15.710
Now you could use
the distance formula,

00:02:15.710 --> 00:02:17.300
but I like to just
cut the chase.

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The distance formula is based
on the Pythagorean theorem,

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and they've drawn a grid
right over here for us,

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so we could just set up some
right triangles to figure out,

00:02:24.736 --> 00:02:26.110
we could make each
of these sides

00:02:26.110 --> 00:02:29.340
essentially the hypotenuse
of right triangles.

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So to figure out
the length of KJ,

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we could set up this right
triangle right over here.

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This height is 3, this
length right over here

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we could count it, it's 1, 2,
3, 4, 5, 6, 7, 8, 9 units wide.

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If we couldn't
count like that, we

00:02:49.210 --> 00:02:51.200
could say, look,
we're going from x

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equals negative 6 to x
equals 3, 3 minus negative 6

00:02:56.360 --> 00:02:57.550
is equal to 9.

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Let me make that clear.

00:02:58.730 --> 00:03:01.271
And that's essentially where
the distance formula comes from,

00:03:01.271 --> 00:03:02.460
3 minus negative 6.

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You're finding the change,
your change in x and you're

00:03:05.670 --> 00:03:08.035
going to square it, and then
you have your change in y,

00:03:08.035 --> 00:03:09.576
you're going to
square it, and that's

00:03:09.576 --> 00:03:11.510
going to be equal to
this hypotenuse squared.

00:03:11.510 --> 00:03:14.500
So let's think
about-- so this is

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going to be equal
to the square root.

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I'm just using the
Pythagorean theorem here.

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So [? that line, ?]
I'll just call it so K,

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let me just write it out,
so that you don't think

00:03:23.400 --> 00:03:25.200
I'm doing some type of voodoo.

00:03:25.200 --> 00:03:27.634
So 3 squared plus-- let
me do it a little higher,

00:03:27.634 --> 00:03:29.050
because I have a
feeling I'm going

00:03:29.050 --> 00:03:33.260
to use a lot of real estate--
3 squared plus 9 squared

00:03:33.260 --> 00:03:39.360
is equal to the
length of KJ squared.

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3 squared is 27, sorry,
not 27, 3 squared is 9.

00:03:44.986 --> 00:03:48.170
9 squared is 81, and
then that's going

00:03:48.170 --> 00:03:51.070
to be equal to the
length of KJ squared.

00:03:51.070 --> 00:03:57.240
Or, we get KJ squared
is equal to 90,

00:03:57.240 --> 00:04:01.710
which is the same
thing as 9 times 10.

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And so KJ is going to
be-- it's the square root

00:04:05.440 --> 00:04:07.300
of this, so the
square root of that

00:04:07.300 --> 00:04:10.590
is going to be the
square root of 9 times 10

00:04:10.590 --> 00:04:13.810
is equal to the
square root of 9 times

00:04:13.810 --> 00:04:18.890
the square root of 10, which is
equal to 3 square roots of 10.

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So it's actually the
exact same length as BA.

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Now let's do the
same thing for KL.

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For KL we could set up another
right triangle right over here.

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So Its height is 3
and its width is 3.

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This is 3 and 3 right over here.

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So we could write 3
squared plus 3 squared

00:04:40.600 --> 00:04:44.960
is equal to the
length of KL squared.

00:04:44.960 --> 00:04:50.500
This right over here is
18 is equal to KL squared,

00:04:50.500 --> 00:04:53.540
and if you take the
square root of both sides

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18 is, of course, 9 times 2.

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So this right over here,
this is the square root of 18

00:04:59.830 --> 00:05:02.510
is equal to KL.

00:05:02.510 --> 00:05:04.060
Square root of 18
is the same thing

00:05:04.060 --> 00:05:05.590
as square root of
9 times 2, that's

00:05:05.590 --> 00:05:06.920
just 3 square roots of 2.

00:05:06.920 --> 00:05:11.030
This is equal to 3
square roots of 2.

00:05:11.030 --> 00:05:14.080
So this is 3 square roots of 2.

00:05:14.080 --> 00:05:18.010
This is 3 square roots of 10.

00:05:18.010 --> 00:05:21.920
This angle is congruent to
this angle right over here.

00:05:21.920 --> 00:05:27.050
So not only are ABC and JKL,
not only are they similar,

00:05:27.050 --> 00:05:29.250
but these are actually
congruent triangles.

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These are congruent
triangles right over here,

00:05:31.166 --> 00:05:34.400
but all congruent triangles
are going to be similar.

00:05:34.400 --> 00:05:38.310
So this one this one
is definitely similar.

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JKL is similar to ABC.

00:05:40.860 --> 00:05:43.440
Now let's figure out all
of these lengths up here.

00:05:43.440 --> 00:05:46.397
Well you have,
let's see, this is 1

00:05:46.397 --> 00:05:51.500
and 1, so you could say
1 squared plus 1 squared

00:05:51.500 --> 00:05:58.470
is equal to EF squared, is equal
to the length of segment EF

00:05:58.470 --> 00:05:59.400
squared.

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Or you could say EF is equal
to-- Let's see, this is 2,

00:06:05.320 --> 00:06:08.851
so this is going to be
the square root of 2.

00:06:08.851 --> 00:06:11.100
So actually I should have
done that in that blue color

00:06:11.100 --> 00:06:13.370
because this will
be the shorter,

00:06:13.370 --> 00:06:16.960
we want this to be the shorter
side if we're comparing the two

00:06:16.960 --> 00:06:19.124
sides on either
side of that angle.

00:06:19.124 --> 00:06:20.165
And now let's look at ED.

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So once again, we can
set up a right triangle

00:06:27.100 --> 00:06:30.570
that is 1 wide and 3 tall.

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And so we could say
1 squared plus 3

00:06:36.170 --> 00:06:41.980
squared is equal to the
length of ED squared.

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1 squared plus 3 squared,
this is going to be 10.

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So we get ED is equal to
the square root of 10.

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Now is the ratio between BC
and EF the same as the ratio

00:06:54.940 --> 00:06:58.430
between BA and ED?

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So let's write this down.

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So square root of 3,
or 3 square roots of 2

00:07:05.070 --> 00:07:08.990
over square root
of 2 is equal to 3.

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So the ratio between these two
blue sides between BC and EF

00:07:12.920 --> 00:07:13.880
is 3.

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The ratio between
these two orange sides

00:07:16.310 --> 00:07:25.410
is 3 square roots of 10
over the square root of 10,

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which is also 3.

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So the ratio between these
two corresponding sides

00:07:29.710 --> 00:07:32.440
are the same, the angle
in between is congruent,

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so these two triangles
are also similar.

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So DEF is also similar to ABC.

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So we have one more
to think about.

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So HI is a candidate to
be corresponding to BC.

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What's its length?

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Well, let's see, this
is a right triangle

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that's width 2, height 2, so
we could write 2 squared plus 2

00:07:57.760 --> 00:08:03.620
squared is equal to the
length of HI squared.

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This is 4, this is 4, so
8 is equal to the length

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of HI squared.

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And then we can take the
square root of both sides.

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So the square root of 8 is
equal to the length of HI,

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or we could say the
length of HI is equal to,

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let's see this is 4 times
2, so the square root

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of 4 times the square root
of 2 is 2 square roots of 2.

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So this is 2 square roots of 2.

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And then the candidate to be
corresponding to AB is HG,

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it's the longer of the two sides
on either side of this angle.

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So HG, we can set up a right
triangle that looks like this.

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It's 2 wide and 1,
2, 3, 4, 5, 6 high.

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So we can write, and
let's scroll over

00:08:56.380 --> 00:08:57.910
to the right a
little bit, we can

00:08:57.910 --> 00:09:01.550
write that 2 squared
plus 6 squared

00:09:01.550 --> 00:09:06.350
is equal to the
length of HG squared.

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So let's see, this
is 4 plus 36 is

00:09:09.900 --> 00:09:13.030
equal to the length
of HG squared.

00:09:13.030 --> 00:09:18.670
Or we get that HG is equal
to the square root of 40.

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Square root of 40, let's see,
how can we simplify that?

00:09:22.786 --> 00:09:26.670
That's the same thing as the
square root of 4 times 10,

00:09:26.670 --> 00:09:29.720
which is 2 times the
square root of 10.

00:09:29.720 --> 00:09:35.280
So HG is equal to 2 times
the square root of 10.

00:09:35.280 --> 00:09:37.000
Now let's compare the ratios.

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The ratio between
BC and HI is going

00:09:40.650 --> 00:09:51.030
to be 3 square roots of 2 over
2 square roots of 2, which

00:09:51.030 --> 00:09:53.730
is 3/2.

00:09:53.730 --> 00:09:59.820
And the ratio
between BA-- let me

00:09:59.820 --> 00:10:06.460
do that in that orange color--
which is 3 square roots of 10,

00:10:06.460 --> 00:10:12.990
and HG, which is 2 square
roots of 10, is also 3/2.

00:10:12.990 --> 00:10:16.135
So all of these triangles, not
only are they similar to ABC,

00:10:16.135 --> 00:10:20.950
but they're actually all
similar to each other.


Classifying a quadrilateral on the coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.770
Classify quadrilateral ABCD.

00:00:03.770 --> 00:00:06.380
Choose the option that best
suits the quadrilateral.

00:00:06.380 --> 00:00:09.530
We're going to pick whether it's
a square, rhombus, rectangle,

00:00:09.530 --> 00:00:12.710
parallelogram, trapezoid,
none of the above.

00:00:12.710 --> 00:00:14.920
And I'm assuming we're
going to pick the most

00:00:14.920 --> 00:00:19.000
specific one possible, because
obviously all squares are

00:00:19.000 --> 00:00:21.670
rhombuses, or rhombi,
I guess you'd say.

00:00:21.670 --> 00:00:22.900
Not all rhombi are squares.

00:00:22.900 --> 00:00:24.910
All squares are also rectangles.

00:00:24.910 --> 00:00:28.370
All squares, rhombi, and
rectangles are parallelograms.

00:00:28.370 --> 00:00:31.720
So we want to be as specific
as possible in picking this.

00:00:31.720 --> 00:00:36.492
So let's see, point
A is at 1 comma 6.

00:00:36.492 --> 00:00:37.950
I encourage you to
pause this video

00:00:37.950 --> 00:00:40.560
and actually try this on your
own before seeing how I do it,

00:00:40.560 --> 00:00:42.290
but I'll just proceed.

00:00:42.290 --> 00:00:46.030
So that's point A
right over there.

00:00:46.030 --> 00:00:48.520
Point B is at
negative 5 comma 2.

00:00:51.830 --> 00:00:53.440
That's point B.

00:00:53.440 --> 00:00:57.640
Point C is at-- just had
some carbonated water,

00:00:57.640 --> 00:00:58.740
so some air is coming up.

00:00:58.740 --> 00:01:00.970
Point C is at
negative 7 comma 8.

00:01:04.849 --> 00:01:07.920
So that is point C
right over there.

00:01:07.920 --> 00:01:11.450
And then finally, point
D is at 2 comma 11.

00:01:14.010 --> 00:01:16.830
And actually that kind
of goes off the screen.

00:01:16.830 --> 00:01:20.940
This is 10, 11 would
be right like this.

00:01:20.940 --> 00:01:22.840
So that would be 2 comma 11.

00:01:22.840 --> 00:01:24.640
If we were to extend
this, this is 10

00:01:24.640 --> 00:01:26.060
and this is 11 right up here.

00:01:26.060 --> 00:01:27.010
2 comma 11.

00:01:27.010 --> 00:01:29.460
So let's see what this
quadrilateral looks like.

00:01:29.460 --> 00:01:34.170
You have this line right
over here, this line right

00:01:34.170 --> 00:01:39.130
over there, that line
right over there,

00:01:39.130 --> 00:01:48.040
and then you have
this line like this,

00:01:48.040 --> 00:01:50.492
and then you have
this like this.

00:01:50.492 --> 00:01:52.950
So right off the bat, well it's
definitely a quadrilateral.

00:01:52.950 --> 00:01:54.440
I have four sides.

00:01:54.440 --> 00:01:57.340
But the key question, are
any of these sides parallel

00:01:57.340 --> 00:01:59.420
to any of the other sides?

00:01:59.420 --> 00:02:08.366
So just looking at it, side CB
is clearly not parallel to AD.

00:02:08.366 --> 00:02:10.270
You just can look at it.

00:02:10.270 --> 00:02:14.950
Now it also looks like
CD is not parallel to BA,

00:02:14.950 --> 00:02:16.760
but maybe I just drew it badly.

00:02:16.760 --> 00:02:18.980
Maybe they actually
are parallel.

00:02:18.980 --> 00:02:21.847
So let's see if we
can verify that.

00:02:21.847 --> 00:02:23.930
So the way to tell whether
two things are parallel

00:02:23.930 --> 00:02:26.790
is to actually figure
out their slopes.

00:02:26.790 --> 00:02:34.460
So let's first figure out
the slope of AB, or BA.

00:02:34.460 --> 00:02:36.530
So let's figure
out our slope here.

00:02:36.530 --> 00:02:39.020
Your slope is going
to be the change

00:02:39.020 --> 00:02:44.350
in y over your change in x.

00:02:44.350 --> 00:02:46.260
And in this case,
you could think of it

00:02:46.260 --> 00:02:51.210
as we're starting at the
point negative 5 comma 2,

00:02:51.210 --> 00:02:54.210
and we're ending at
the point 1 comma 6.

00:02:54.210 --> 00:02:56.710
So what's our change in y?

00:02:56.710 --> 00:02:59.800
Our change in y, we
go from 2-- we're

00:02:59.800 --> 00:03:02.450
going from 2 all the way to 6.

00:03:02.450 --> 00:03:06.880
Or you could say it's 6 minus
2, our change in y is 4.

00:03:06.880 --> 00:03:08.540
What's our change in x?

00:03:08.540 --> 00:03:14.510
Well we go from negative 5
to 1, so we increased by 6.

00:03:14.510 --> 00:03:18.090
Or another way of thinking
about it, 1 minus negative 5,

00:03:18.090 --> 00:03:19.580
well that's going
to be equal to 6.

00:03:19.580 --> 00:03:21.590
So our slope here is 2/3.

00:03:21.590 --> 00:03:23.990
It's 2/3.

00:03:23.990 --> 00:03:27.160
Every time we move 3
in the x direction,

00:03:27.160 --> 00:03:28.960
we go up 2 in the y direction.

00:03:28.960 --> 00:03:32.200
Move 3 in the x direction,
go up 2 in the y direction.

00:03:32.200 --> 00:03:34.410
Now let's think about
line CD up here.

00:03:38.260 --> 00:03:39.620
What is our slope?

00:03:39.620 --> 00:03:46.180
Our change in y over change
in x is going to be equal to,

00:03:46.180 --> 00:03:46.690
let's see.

00:03:46.690 --> 00:03:49.940
Our change in-- let's figure
out our change in x first.

00:03:49.940 --> 00:03:56.070
Our change in x, we're
going from negative 7

00:03:56.070 --> 00:04:00.250
comma 8 all the
way to 2 comma 11.

00:04:00.250 --> 00:04:05.270
So our change in x, we're
going from negative 7 to 2.

00:04:05.270 --> 00:04:08.460
Or we could say 2
minus negative 7.

00:04:08.460 --> 00:04:11.930
So we are increasing by 9.

00:04:11.930 --> 00:04:13.620
That's our change in x.

00:04:13.620 --> 00:04:15.720
So that's going
to be equal to 9.

00:04:15.720 --> 00:04:19.029
And then our change in y, well
it looks like we've gone up.

00:04:19.029 --> 00:04:22.910
We've gone from 8 to
11, so we've gone up 3.

00:04:22.910 --> 00:04:25.870
Or we could say 11 minus 8.

00:04:25.870 --> 00:04:28.150
Notice, endpoint
minus start point.

00:04:28.150 --> 00:04:29.500
Endpoint minus start point.

00:04:29.500 --> 00:04:32.490
You have to do that on the
top and the bottom, otherwise

00:04:32.490 --> 00:04:34.990
you're not going to actually
be calculating your change in y

00:04:34.990 --> 00:04:36.130
over change in x.

00:04:36.130 --> 00:04:40.690
But you notice our change,
when our x increases by 9,

00:04:40.690 --> 00:04:43.630
our y increased by 3.

00:04:43.630 --> 00:04:46.640
So the slope here
is equal to 1/3.

00:04:46.640 --> 00:04:49.010
So these actually
have different slopes.

00:04:49.010 --> 00:04:51.410
So none of these lines are
parallel to each other.

00:04:51.410 --> 00:04:53.190
So this isn't even
a parallelogram.

00:04:53.190 --> 00:04:54.390
This isn't even a trapezoid.

00:04:54.390 --> 00:04:56.950
Parallelogram, you have to have
two pairs of parallel sides.

00:04:56.950 --> 00:04:59.530
Trapezoid, you have to have
one pair of parallel sides.

00:04:59.530 --> 00:05:02.520
This isn't the case for any of
these, or none of these sides

00:05:02.520 --> 00:05:03.370
are parallel.

00:05:03.370 --> 00:05:07.395
So we would go with
none of the above.


Recognizing points on a circle

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.160
Point A is at
negative 5 comma 5.

00:00:03.160 --> 00:00:05.230
So this is negative
5 right over here.

00:00:05.230 --> 00:00:06.900
This is 1, 2, 3, 4, 5.

00:00:06.900 --> 00:00:08.430
That's 5 right over there.

00:00:08.430 --> 00:00:12.990
So point A is right about there.

00:00:12.990 --> 00:00:18.550
So that is point A, just like
that, at negative 5 comma 5.

00:00:18.550 --> 00:00:20.340
And then, it's a center
of circle A, which

00:00:20.340 --> 00:00:23.006
I won't draw just yet, because I
don't know the radius of circle

00:00:23.006 --> 00:00:26.963
A. Point B is at--
let me underline these

00:00:26.963 --> 00:00:30.880
in the appropriate color--
point B is at 3 comma 1.

00:00:30.880 --> 00:00:32.590
So 1, 2, 3 comma 1.

00:00:32.590 --> 00:00:35.320
So that's point B
right over there,

00:00:35.320 --> 00:00:40.510
so center of circle
B. Point P is at 0, 0,

00:00:40.510 --> 00:00:42.350
so it's right over
there at the origin.

00:00:42.350 --> 00:00:44.760
And it is on circles A and B.

00:00:44.760 --> 00:00:46.510
Well, that's a big
piece of information.

00:00:46.510 --> 00:00:50.950
Because that tells us, if
this is on both circles,

00:00:50.950 --> 00:00:53.500
then that means that
this is B's radius

00:00:53.500 --> 00:00:56.460
away from the point
B from the center.

00:00:56.460 --> 00:00:59.350
And this tells us that it
is circle A's radius away

00:00:59.350 --> 00:01:01.130
from its center,
which is at point A.

00:01:01.130 --> 00:01:05.459
So let's figure out what
those radii actually are.

00:01:05.459 --> 00:01:09.360
And so we can
imagine-- let me draw

00:01:09.360 --> 00:01:11.610
a radius or the
radius for circle

00:01:11.610 --> 00:01:14.330
A-- we now know
since P sits on it,

00:01:14.330 --> 00:01:21.217
that this could be considered
the radius for circle A.

00:01:21.217 --> 00:01:23.550
And you could use a distance
formula, but what we'll see

00:01:23.550 --> 00:01:25.341
is that the distance
formula is really just

00:01:25.341 --> 00:01:27.110
falling out of the
Pythagorean theorem.

00:01:27.110 --> 00:01:29.650
So the distance formula tells
us the radius right over here,

00:01:29.650 --> 00:01:31.900
this is just the distance
between those two points.

00:01:31.900 --> 00:01:34.850
So the radius or the distance
between those two points

00:01:34.850 --> 00:01:39.570
squared is going to be equal
to our change in x values

00:01:39.570 --> 00:01:44.100
between A and P. So
our change in x values,

00:01:44.100 --> 00:01:48.820
we could write it as
negative 5 minus 0 squared.

00:01:51.500 --> 00:01:58.060
That's our change in x,
negative 5 minus 0 squared,

00:01:58.060 --> 00:02:06.130
plus our change in
y, 5 minus 0 squared,

00:02:06.130 --> 00:02:10.539
which gets us that our distance
between these two points,

00:02:10.539 --> 00:02:13.560
which is the length
of the radius squared,

00:02:13.560 --> 00:02:20.880
is equal to negative 5
squared plus 5 squared.

00:02:20.880 --> 00:02:23.280
Or we could say
that the radius is

00:02:23.280 --> 00:02:27.990
equal to the square root of,
this is 25, this is 25, 50.

00:02:27.990 --> 00:02:32.072
50, we can write as 25 times 2.

00:02:32.072 --> 00:02:33.530
So this is equal
to the square root

00:02:33.530 --> 00:02:35.210
of 25 times the square
root of 2, which

00:02:35.210 --> 00:02:38.160
is 5 times the square root of 2.

00:02:38.160 --> 00:02:41.010
So this distance right over
here is 5 times the square root

00:02:41.010 --> 00:02:41.770
of 2.

00:02:41.770 --> 00:02:43.730
Now, I said this is just the
same thing as the Pythagorean

00:02:43.730 --> 00:02:44.229
theorem.

00:02:44.229 --> 00:02:45.040
Why?

00:02:45.040 --> 00:02:51.780
Well, if we were to construct a
right triangle right over here,

00:02:51.780 --> 00:02:53.320
then we can look
at this distance.

00:02:53.320 --> 00:02:56.210
This distance would be the
absolute value of negative 5

00:02:56.210 --> 00:02:56.980
minus 0.

00:02:56.980 --> 00:02:59.100
Or you could say, this
is 0 minus negative 5.

00:02:59.100 --> 00:03:01.200
This distance right
over here is 5.

00:03:01.200 --> 00:03:04.030
This distance is the
distance between 0 and 5

00:03:04.030 --> 00:03:04.840
in the y direction.

00:03:04.840 --> 00:03:05.920
That's 5.

00:03:05.920 --> 00:03:08.180
Pythagorean theorem tells
us that 5 squared, which

00:03:08.180 --> 00:03:11.710
is 25, plus 5
squared, another 25,

00:03:11.710 --> 00:03:14.090
is going to be equal to
your hypotenuse squared.

00:03:14.090 --> 00:03:15.659
And that's exactly
what we have here.

00:03:15.659 --> 00:03:17.450
Now, you might be
saying, wait, wait, wait,

00:03:17.450 --> 00:03:19.520
this thing had a
negative 5 squared here,

00:03:19.520 --> 00:03:20.924
while here, you
had a positive 5.

00:03:20.924 --> 00:03:23.340
But the reason why we could
do this is when you square it,

00:03:23.340 --> 00:03:24.624
the negative disappears.

00:03:24.624 --> 00:03:26.290
The distance formula,
you could write it

00:03:26.290 --> 00:03:28.410
this way, where you're
taking the absolute value.

00:03:28.410 --> 00:03:30.410
And then, it becomes very
clear that this really

00:03:30.410 --> 00:03:31.860
is just the Pythagorean theorem.

00:03:31.860 --> 00:03:34.260
This would be 5
squared plus 5 squared.

00:03:34.260 --> 00:03:35.844
5 squared plus 5 squared.

00:03:35.844 --> 00:03:37.510
The reason why you
don't have to do this

00:03:37.510 --> 00:03:39.700
is because a sign doesn't
matter when you square it.

00:03:39.700 --> 00:03:42.760
It always will become
a positive value.

00:03:42.760 --> 00:03:45.360
But either way, we
figured out this radius.

00:03:45.360 --> 00:03:54.940
Now, let's figure out the radius
of circle B. Same exact idea.

00:03:54.940 --> 00:04:02.320
The radius of circle B squared
is equal to our change in x.

00:04:02.320 --> 00:04:07.940
So we could write as 3
minus 0, or 0 minus 3.

00:04:07.940 --> 00:04:10.310
But we'll just write
it as 3 minus 0.

00:04:10.310 --> 00:04:16.290
3 minus 0 squared plus
1 minus 0 squared.

00:04:16.290 --> 00:04:20.430
Or the radius or the distance
between these two points

00:04:20.430 --> 00:04:22.350
is equal to the square
root of-- let's see,

00:04:22.350 --> 00:04:23.980
this is 3 squared
plus 1 squared.

00:04:23.980 --> 00:04:25.340
That's 9 plus 1.

00:04:25.340 --> 00:04:28.270
This is the square root of 10.

00:04:28.270 --> 00:04:32.242
The radius of B is
the square root of 10.

00:04:32.242 --> 00:04:34.200
Now, they ask us, which
of the following points

00:04:34.200 --> 00:04:36.320
are on circle A, circle
B, or both circles?

00:04:36.320 --> 00:04:39.280
So all we have to do now
is look at these points.

00:04:39.280 --> 00:04:43.190
If this point is the square
root of 10 away from point B,

00:04:43.190 --> 00:04:44.235
then it's on the circle.

00:04:44.235 --> 00:04:45.670
It's a radius away.

00:04:45.670 --> 00:04:47.910
A circle is the locus
of all points that

00:04:47.910 --> 00:04:49.940
are a radius away
from the center.

00:04:49.940 --> 00:04:52.570
If it's 5 square roots
of 2 from this point,

00:04:52.570 --> 00:04:55.780
then it's on circle A. If it's
neither, then it's neither.

00:04:55.780 --> 00:04:57.210
Or it could be both.

00:04:57.210 --> 00:04:59.130
So let's try these
out, one by one.

00:04:59.130 --> 00:05:01.930
So point C is at 4, negative 2.

00:05:01.930 --> 00:05:04.100
So let me color
this in a new color.

00:05:04.100 --> 00:05:07.470
So point C-- let me do
it in orange-- point

00:05:07.470 --> 00:05:11.240
C is at 1, 2, 3, 4, negative 2.

00:05:11.240 --> 00:05:12.880
Point C is right over there.

00:05:12.880 --> 00:05:14.540
Now, it looks pretty close.

00:05:14.540 --> 00:05:18.410
Just, this is a hand-drawn
drawing, so it's not perfect.

00:05:18.410 --> 00:05:19.290
So point C is there.

00:05:19.290 --> 00:05:20.300
It looks pretty close.

00:05:20.300 --> 00:05:23.930
But let's actually verify it.

00:05:23.930 --> 00:05:27.070
The distance between
point C and point D,

00:05:27.070 --> 00:05:30.860
so the distance squared, is
going to be equal to the change

00:05:30.860 --> 00:05:31.700
in x's.

00:05:31.700 --> 00:05:36.080
So we could say, 4 minus--
so we're trying [INAUDIBLE]

00:05:36.080 --> 00:05:46.930
C and B, is 4 minus 3 squared
plus negative 2 minus 1

00:05:46.930 --> 00:05:54.210
squared, which is equal
to-- This is 1 squared

00:05:54.210 --> 00:05:57.920
plus negative 3 squared.

00:05:57.920 --> 00:06:03.260
And so our distance
squared is equal to 10.

00:06:03.260 --> 00:06:06.100
Or our distance is equal
to the square root of 10.

00:06:06.100 --> 00:06:08.159
So this is also, the
distance right over here

00:06:08.159 --> 00:06:09.200
is the square root of 10.

00:06:09.200 --> 00:06:10.600
So this is on the circle.

00:06:10.600 --> 00:06:12.850
If we wanted to
draw circle B, it

00:06:12.850 --> 00:06:15.530
would look something like this.

00:06:15.530 --> 00:06:18.060
And once again, I'm hand-drawing
it, so it's not perfect.

00:06:18.060 --> 00:06:19.435
But it would look
something-- I'm

00:06:19.435 --> 00:06:21.840
going to draw part of it--
it looks something like this.

00:06:21.840 --> 00:06:23.770
This is exactly a radius away.

00:06:23.770 --> 00:06:30.560
So let me write, this
is on circle B. Now,

00:06:30.560 --> 00:06:31.610
let's look at this point.

00:06:31.610 --> 00:06:33.790
The point 5 comma 3.

00:06:33.790 --> 00:06:37.480
So I'll do that in pink.

00:06:37.480 --> 00:06:41.540
So 1, 2, 3, 4, 5 comma 3.

00:06:41.540 --> 00:06:45.800
So this looks close, but
let's verify, just in case.

00:06:45.800 --> 00:06:48.614
So now, our distance is equal
to-- let me just write it

00:06:48.614 --> 00:06:50.030
this way-- our
distance squared is

00:06:50.030 --> 00:06:51.760
going to be our
change in x squared.

00:06:51.760 --> 00:07:00.302
So 5 minus 3 squared plus 3
minus 1 squared, change in y.

00:07:03.170 --> 00:07:06.030
And so our distance is going
to be equal to-- actually

00:07:06.030 --> 00:07:07.530
I don't want to
skip too many steps.

00:07:07.530 --> 00:07:09.440
Let's see, this is
2 squared, which

00:07:09.440 --> 00:07:13.420
is 4, plus 2 squared,
which is another 4.

00:07:13.420 --> 00:07:18.700
So our distance is going to be
equal to the square root of 8,

00:07:18.700 --> 00:07:21.720
which is the same thing as
the square root of 2 times 4,

00:07:21.720 --> 00:07:26.140
which is the same thing as 2
times the square root of 2.

00:07:26.140 --> 00:07:27.750
Square root of 4 is 2.

00:07:27.750 --> 00:07:30.350
And then, of course, you just
have the 2 left in the radical.

00:07:30.350 --> 00:07:34.070
So this is a different distance
away than square root of 10.

00:07:34.070 --> 00:07:37.670
So this one right over here
is definitely not on circle B.

00:07:37.670 --> 00:07:39.340
And just eyeballing
it, you can see

00:07:39.340 --> 00:07:42.180
that it's not going to be on
circle A. This distance, just

00:07:42.180 --> 00:07:47.400
eyeballing it, is much further
than 5 square roots of 2.

00:07:47.400 --> 00:07:49.410
And that's also true
for point C. Point C

00:07:49.410 --> 00:07:51.190
is much further than
5 square roots of 2.

00:07:51.190 --> 00:07:54.130
You can just look
at that visually.

00:07:54.130 --> 00:07:56.700
They're much further than
a radius away from A.

00:07:56.700 --> 00:08:00.970
So this point right over
here, this is neither.

00:08:00.970 --> 00:08:03.420
This is on neither circle.

00:08:03.420 --> 00:08:06.160
Now, finally, we have the
point, negative 2 comma 8.

00:08:06.160 --> 00:08:09.030
So let me find-- I'm
running out of colors.

00:08:09.030 --> 00:08:12.790
Let me see, I could use, so I
guess I'll use yellow again.

00:08:12.790 --> 00:08:17.628
Negative 2 comma 8, so that's
negative 2 comma 1, 2, 3, 4, 5,

00:08:17.628 --> 00:08:19.000
6, 7, 8.

00:08:19.000 --> 00:08:23.300
So it's right over here.

00:08:23.300 --> 00:08:27.130
That is point E.
Just eyeballing it,

00:08:27.130 --> 00:08:30.200
this distance-- so it's
clearly way too far.

00:08:30.200 --> 00:08:31.860
Just looking at,
just eyeballing it,

00:08:31.860 --> 00:08:34.260
it's clearly more than
a radius away from B.

00:08:34.260 --> 00:08:36.330
So this isn't going
to be on circle B.

00:08:36.330 --> 00:08:39.250
And also, looking at
it relative to point A,

00:08:39.250 --> 00:08:43.539
it looks much closer to point
A. It doesn't even seem close,

00:08:43.539 --> 00:08:44.880
than point P is.

00:08:44.880 --> 00:08:47.650
So it looks, just inspecting
it, that you could rule this one

00:08:47.650 --> 00:08:50.060
out, that this is
going to be neither.

00:08:50.060 --> 00:08:52.380
But we can verify this
on our own, if we like.

00:08:52.380 --> 00:08:55.020
We can just find the distance
between these two points.

00:08:55.020 --> 00:08:58.100
Our distance squared is going
to be our change in x's.

00:08:58.100 --> 00:09:08.590
So negative 2 minus negative 5
squared plus our change in y.

00:09:08.590 --> 00:09:13.040
So it's 8 minus 5 squared.

00:09:13.040 --> 00:09:14.970
And so this is, our
distance squared

00:09:14.970 --> 00:09:18.520
is going to be equal to
negative 2 minus negative 5.

00:09:18.520 --> 00:09:20.840
That's negative 2 plus 5.

00:09:20.840 --> 00:09:24.671
So that's going to be 3
squared plus 3 squared.

00:09:24.671 --> 00:09:26.920
And you see that right over
here, Pythagorean theorem.

00:09:26.920 --> 00:09:29.570
This distance right
over here is 3.

00:09:29.570 --> 00:09:32.760
This distance right over here,
this is your change in x is 3.

00:09:32.760 --> 00:09:34.000
Change in y is 3.

00:09:34.000 --> 00:09:36.290
3 squared plus 3
squared is going

00:09:36.290 --> 00:09:38.870
to be the distance squared,
the hypotenuse squared.

00:09:38.870 --> 00:09:42.530
So our distance squared is
going to be-- or I could say,

00:09:42.530 --> 00:09:44.280
our distance--
skip a few steps--

00:09:44.280 --> 00:09:48.850
is equal to the square root of--
we can write this as 9 times 2

00:09:48.850 --> 00:09:52.940
or the distance is equal to
3 times the square root of 2.

00:09:52.940 --> 00:09:55.286
The radius of circle A is 5
times the square root of 2,

00:09:55.286 --> 00:09:56.660
not 3 times the
square root of 2.

00:09:56.660 --> 00:09:59.960
So this is actually going
to be inside the circle.

00:09:59.960 --> 00:10:01.840
So if we want to
draw circle A, it's

00:10:01.840 --> 00:10:05.390
going to look
something like this.

00:10:07.950 --> 00:10:11.730
And point E is on the inside.

00:10:11.730 --> 00:10:15.250
Point D and point C are on
the outside of circle A.

00:10:15.250 --> 00:10:19.920
The only one that sits on any
of the circles is point C.


Similarity to define sine, cosine, and tangent

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.110
We've got two right
triangles here.

00:00:03.110 --> 00:00:05.670
And let's say we also
know that they both have

00:00:05.670 --> 00:00:09.810
an angle whose measure
is equal to theta.

00:00:09.810 --> 00:00:13.440
So angle A is congruent
to angle D. What

00:00:13.440 --> 00:00:16.840
do we now know about
these two triangles?

00:00:16.840 --> 00:00:20.080
Well for any triangle, if
you know two of the angles,

00:00:20.080 --> 00:00:21.980
you're going to know
the third angle,

00:00:21.980 --> 00:00:24.290
because the sum of the
angles of a triangle

00:00:24.290 --> 00:00:26.440
add up to 180 degrees.

00:00:26.440 --> 00:00:27.940
So if you have two
angles in common,

00:00:27.940 --> 00:00:30.320
that means you're going to
have three angles in common.

00:00:30.320 --> 00:00:32.170
And if you have three
angles in common,

00:00:32.170 --> 00:00:35.010
you are dealing with
similar triangles.

00:00:35.010 --> 00:00:37.130
Let me make that a
little bit clearer.

00:00:37.130 --> 00:00:39.680
So if this angle is
theta, this is 90.

00:00:39.680 --> 00:00:41.570
They all have to add
up to 180 degrees.

00:00:41.570 --> 00:00:43.570
That means that this
angle plus this angle

00:00:43.570 --> 00:00:44.790
up here have to add up to 90.

00:00:44.790 --> 00:00:46.930
We've already used up
90 right over here,

00:00:46.930 --> 00:00:50.710
so angle A and angle B
need to be complements.

00:00:50.710 --> 00:00:54.490
So this angle right over here
needs to be 90 minus theta.

00:00:54.490 --> 00:00:56.620
Well we could use the
same logic over here.

00:00:56.620 --> 00:00:58.970
We already use of 90
degrees over here.

00:00:58.970 --> 00:01:01.120
So we have a
remaining 90 degrees

00:01:01.120 --> 00:01:02.860
between theta and that angle.

00:01:02.860 --> 00:01:06.785
So this angle is going to
be 90 degrees minus theta.

00:01:10.280 --> 00:01:14.400
You have three corresponding
angles being congruent.

00:01:14.400 --> 00:01:16.970
You are dealing with
similar triangles.

00:01:16.970 --> 00:01:19.090
Now why is that interesting?

00:01:19.090 --> 00:01:21.590
Well we know from
geometry that the ratio

00:01:21.590 --> 00:01:24.010
of corresponding sides
of similar triangles

00:01:24.010 --> 00:01:25.870
are always going to be the same.

00:01:25.870 --> 00:01:28.660
So let's explore the
corresponding sides here.

00:01:28.660 --> 00:01:30.897
Well, the side that
jumps out-- when

00:01:30.897 --> 00:01:32.980
you're dealing with the
right triangles-- the most

00:01:32.980 --> 00:01:34.270
is always the hypotenuse.

00:01:34.270 --> 00:01:36.810
So this right over
here is the hypotenuse.

00:01:36.810 --> 00:01:38.550
This hypotenuse is
going to correspond

00:01:38.550 --> 00:01:41.210
to this hypotenuse
right over here.

00:01:41.210 --> 00:01:43.160
And then we could
write that down.

00:01:43.160 --> 00:01:45.410
This is the hypotenuse
of this triangle.

00:01:45.410 --> 00:01:49.730
This is the hypotenuse
of that triangle.

00:01:49.730 --> 00:01:52.410
Now this side right
over here, side BC,

00:01:52.410 --> 00:01:54.504
what side does
that correspond to?

00:01:54.504 --> 00:01:55.920
Well if you look
at this triangle,

00:01:55.920 --> 00:02:00.430
you can view it as the side that
is opposite this angle theta.

00:02:00.430 --> 00:02:01.145
So it's opposite.

00:02:01.145 --> 00:02:03.020
If you go across the
triangle, you get there.

00:02:03.020 --> 00:02:04.540
So let's go opposite angle D.

00:02:04.540 --> 00:02:06.610
If you go opposite
angle A, you get to BC.

00:02:06.610 --> 00:02:11.110
Opposite angle D, you get to EF.

00:02:11.110 --> 00:02:14.150
So it corresponds to this
side right over here.

00:02:14.150 --> 00:02:17.510
And then finally, side AC
is the one remaining one.

00:02:17.510 --> 00:02:19.340
We could view it
as, well, there's

00:02:19.340 --> 00:02:22.280
two sides that make up
this angle A. One of them

00:02:22.280 --> 00:02:23.210
is the hypotenuse.

00:02:23.210 --> 00:02:25.900
We could call this, maybe,
the adjacent side to it.

00:02:25.900 --> 00:02:28.990
And so D corresponds
to A, and so this

00:02:28.990 --> 00:02:31.500
would be the side
that corresponds.

00:02:31.500 --> 00:02:33.650
Now the whole reason I
did that is to leverage

00:02:33.650 --> 00:02:35.500
that, corresponding
sides, the ratio

00:02:35.500 --> 00:02:37.550
between corresponding
sides of similar triangles,

00:02:37.550 --> 00:02:40.140
is always going to be the same.

00:02:40.140 --> 00:02:44.820
So for example, the ratio
between BC and the hypotenuse,

00:02:44.820 --> 00:02:53.830
BA-- so let me write
that down-- BC/BA

00:02:53.830 --> 00:03:04.660
is going to be equal to EF/ED,
the length of segment EF

00:03:04.660 --> 00:03:07.020
over the length of segment ED.

00:03:12.050 --> 00:03:14.890
Or we could also
write that the length

00:03:14.890 --> 00:03:20.940
of segment AC over
the hypotenuse,

00:03:20.940 --> 00:03:24.760
over this triangle's
hypotenuse, over AB,

00:03:24.760 --> 00:03:31.010
is equal to DF/DE-- once
again, this green side

00:03:31.010 --> 00:03:31.885
over the orange side.

00:03:31.885 --> 00:03:33.051
These are similar triangles.

00:03:33.051 --> 00:03:34.790
They're corresponding
to each other.

00:03:34.790 --> 00:03:38.317
So this is equal to DF/DE.

00:03:45.701 --> 00:03:47.950
And we could keep going, but
I'll just do another one.

00:03:47.950 --> 00:03:50.440
Or we could say that the
ratio of this side right

00:03:50.440 --> 00:03:53.180
over here-- this blue
side to the green side

00:03:53.180 --> 00:04:01.240
of this triangle--
the length of BC/CA

00:04:01.240 --> 00:04:05.000
is going to be the same as
the ratio between these two

00:04:05.000 --> 00:04:08.297
corresponding sides, the
blue over the green, EF/DF.

00:04:15.240 --> 00:04:17.630
And we got all of
this from the fact

00:04:17.630 --> 00:04:19.640
that these are
similar triangles.

00:04:19.640 --> 00:04:25.091
So this is true for
any right triangle

00:04:25.091 --> 00:04:26.090
that has an angle theta.

00:04:26.090 --> 00:04:29.220
Then those two triangles
are going to be similar,

00:04:29.220 --> 00:04:31.720
and all of these ratios
are going to be the same.

00:04:31.720 --> 00:04:34.390
Well, maybe we can give
names to these ratios

00:04:34.390 --> 00:04:38.750
relative to the angle theta.

00:04:38.750 --> 00:04:43.150
So from angle theta's
point of view--

00:04:43.150 --> 00:04:44.840
I'll write theta
right over here,

00:04:44.840 --> 00:04:47.700
or we can just
remember that-- what

00:04:47.700 --> 00:04:49.220
is the ratio of these two sides?

00:04:49.220 --> 00:04:52.720
Well from theta's point
of view, that blue side

00:04:52.720 --> 00:04:54.310
is the opposite side.

00:04:54.310 --> 00:04:57.790
It's opposite-- so the opposite
side of the right triangle.

00:04:57.790 --> 00:05:00.940
And then the orange side we've
already labeled the hypotenuse.

00:05:00.940 --> 00:05:03.140
So from theta's
point of view, this

00:05:03.140 --> 00:05:07.380
is the opposite side
over the hypotenuse.

00:05:07.380 --> 00:05:09.340
And I keep stating from
theta's point of view

00:05:09.340 --> 00:05:12.430
because that wouldn't be the
case for this other angle,

00:05:12.430 --> 00:05:14.990
for angle B. From angle
B's point of view,

00:05:14.990 --> 00:05:18.220
this is the adjacent
side over the hypotenuse.

00:05:18.220 --> 00:05:20.500
And we'll think about that
relationship later on.

00:05:20.500 --> 00:05:22.541
But let's just all think
of it from theta's point

00:05:22.541 --> 00:05:23.720
of view right over here.

00:05:23.720 --> 00:05:26.170
So from theta's point
of view, what is this?

00:05:26.170 --> 00:05:28.060
Well theta's right over here.

00:05:28.060 --> 00:05:34.260
Clearly AB and DE are still
the hypotenuses-- hypoteni.

00:05:34.260 --> 00:05:36.370
I don't know how to say
that in plural again.

00:05:36.370 --> 00:05:40.570
And what is AC, and what are DF?

00:05:40.570 --> 00:05:42.170
Well, these are adjacent to it.

00:05:42.170 --> 00:05:43.990
They're one of
the two sides that

00:05:43.990 --> 00:05:47.210
make up this angle that
is not the hypotenuse.

00:05:47.210 --> 00:05:50.230
So this we can view as
the ratio, in either

00:05:50.230 --> 00:05:53.110
of these triangles,
between the adjacent side--

00:05:53.110 --> 00:05:54.060
so this is relative.

00:05:54.060 --> 00:05:55.792
Once again, this is
opposite angle B,

00:05:55.792 --> 00:05:57.250
but we're only
thinking about angle

00:05:57.250 --> 00:05:59.770
A right here, or the
angle that measures theta,

00:05:59.770 --> 00:06:02.520
or angle D right over
here-- relative to angle A,

00:06:02.520 --> 00:06:03.750
AC is adjacent.

00:06:03.750 --> 00:06:08.060
Relative to angle
D, DF is adjacent.

00:06:08.060 --> 00:06:11.490
So this ratio right over
here is the adjacent

00:06:11.490 --> 00:06:13.350
over the hypotenuse.

00:06:13.350 --> 00:06:17.470
And it's going to be the same
for any right triangle that

00:06:17.470 --> 00:06:20.040
has an angle theta in it.

00:06:20.040 --> 00:06:22.260
And then finally,
this over here, this

00:06:22.260 --> 00:06:23.850
is going to be
the opposite side.

00:06:23.850 --> 00:06:26.060
Once again, this was the
opposite side over here.

00:06:26.060 --> 00:06:27.970
This ratio for
either right triangle

00:06:27.970 --> 00:06:32.410
is going to be the opposite
side over the adjacent side.

00:06:35.157 --> 00:06:36.990
And I really want to
stress the importance--

00:06:36.990 --> 00:06:39.240
and we're going to do many,
many more examples of this

00:06:39.240 --> 00:06:43.040
to make this very concrete--
but for any right triangle

00:06:43.040 --> 00:06:47.620
that has an angle theta, the
ratio between its opposite side

00:06:47.620 --> 00:06:50.130
and its hypotenuse is
going to be the same.

00:06:50.130 --> 00:06:51.790
That comes out of
similar triangles.

00:06:51.790 --> 00:06:52.960
We've just explored that.

00:06:52.960 --> 00:06:55.000
The ratio between
the adjacent side

00:06:55.000 --> 00:06:57.280
to that angle that is
theta and the hypotenuse

00:06:57.280 --> 00:07:00.630
is going to be the same,
for any of these triangles,

00:07:00.630 --> 00:07:02.640
as long as it has that
angle theta in it.

00:07:02.640 --> 00:07:04.940
And the ratio,
relative to the angle

00:07:04.940 --> 00:07:07.560
theta, between the opposite
side and the adjacent side,

00:07:07.560 --> 00:07:09.390
between the blue side
and the green side,

00:07:09.390 --> 00:07:11.340
is always going to be the same.

00:07:11.340 --> 00:07:13.370
These are similar triangles.

00:07:13.370 --> 00:07:15.980
So given that,
mathematicians decided

00:07:15.980 --> 00:07:17.990
to give these things names.

00:07:17.990 --> 00:07:20.060
Relative to the angle
theta, this ratio

00:07:20.060 --> 00:07:23.330
is always going to be
the same, so the opposite

00:07:23.330 --> 00:07:26.755
over hypotenuse, they call this
the sine of the angle theta.

00:07:29.310 --> 00:07:34.065
Let me do this in a new
color-- by definition--

00:07:34.065 --> 00:07:36.440
and we're going to extend this
definition in the future--

00:07:36.440 --> 00:07:39.160
this is sine of theta.

00:07:39.160 --> 00:07:41.840
This right over
here, by definition,

00:07:41.840 --> 00:07:44.720
is the cosine of theta.

00:07:44.720 --> 00:07:47.060
And this right over
here, by definition,

00:07:47.060 --> 00:07:53.720
is the tangent of theta.

00:07:53.720 --> 00:07:56.260
And a mnemonic that will help
you remember this-- and these

00:07:56.260 --> 00:07:57.810
really are just definitions.

00:07:57.810 --> 00:08:00.810
People realized, wow, by similar
triangles, for any angle theta,

00:08:00.810 --> 00:08:03.637
this ratio is always
going to be the same.

00:08:03.637 --> 00:08:05.720
Because of similar triangles,
for any angle theta,

00:08:05.720 --> 00:08:07.470
this ratio is always
going to be the same.

00:08:07.470 --> 00:08:09.170
This ratio is always
going to the same.

00:08:09.170 --> 00:08:11.670
So let's make these definitions.

00:08:11.670 --> 00:08:13.480
And to help us
remember it, there's

00:08:13.480 --> 00:08:15.760
the mnemonic soh-cah-toa.

00:08:15.760 --> 00:08:18.060
So I'll write it like this.

00:08:18.060 --> 00:08:23.340
soh is sine is opposite
over hypotenuse.

00:08:23.340 --> 00:08:27.845
cah-- cosine is adjacent
over hypotenuse.

00:08:32.350 --> 00:08:36.710
And then finally,
tangent is opposite

00:08:36.710 --> 00:08:47.520
over adjacent-- soh-cah-toa.

00:08:47.520 --> 00:08:49.020
And in future videos,
we'll actually

00:08:49.020 --> 00:08:55.276
apply these definitions for
these trigonometric functions.


Showing relationship between cosine and sine of complements

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.460
We see in a triangle, or I
guess we know in a triangle,

00:00:03.460 --> 00:00:04.560
there's three angles.

00:00:04.560 --> 00:00:05.900
And if we're talking
about a right triangle,

00:00:05.900 --> 00:00:07.816
like the one that I've
drawn here, one of them

00:00:07.816 --> 00:00:09.640
is going to be a right angle.

00:00:09.640 --> 00:00:12.450
And so we have two other
angles to deal with.

00:00:12.450 --> 00:00:14.220
And what I want to
explore in this video

00:00:14.220 --> 00:00:18.020
is the relationship between
the sine of one of these angles

00:00:18.020 --> 00:00:20.770
and the cosine of the other, the
cosine of one of these angles

00:00:20.770 --> 00:00:23.240
and the sine of the other.

00:00:23.240 --> 00:00:25.220
So to do that, let's just
say that this angle--

00:00:25.220 --> 00:00:27.330
I guess we could call
it angle A-- let's say

00:00:27.330 --> 00:00:28.780
it's equal to theta.

00:00:28.780 --> 00:00:31.280
If this is equal to
theta, if it's measure is

00:00:31.280 --> 00:00:34.810
equal to theta degrees, say,
what is the measure of angle B

00:00:34.810 --> 00:00:36.920
going to be?

00:00:36.920 --> 00:00:38.790
Well, the thing that
will jump out at you--

00:00:38.790 --> 00:00:41.070
and we've looked at
this in other problems--

00:00:41.070 --> 00:00:42.750
is the sum of the
angles of a triangle

00:00:42.750 --> 00:00:44.480
are going to be 180 degrees.

00:00:44.480 --> 00:00:46.850
And this one right over
here, it's a right triangle.

00:00:46.850 --> 00:00:50.100
So this right angle takes
up 90 of those 180 degrees.

00:00:50.100 --> 00:00:51.720
So you have 90 degrees left.

00:00:51.720 --> 00:00:55.370
So these two are going to
have to add up to 90 degrees.

00:00:55.370 --> 00:00:58.050
This one and this one,
angle A and angle B,

00:00:58.050 --> 00:01:00.150
are going to be
complements of each other.

00:01:00.150 --> 00:01:02.120
They're going to
be complementary.

00:01:02.120 --> 00:01:03.815
Or another way of
thinking about it is B

00:01:03.815 --> 00:01:07.400
could be written
as 90 minus theta.

00:01:07.400 --> 00:01:13.880
If you add theta to 90
degrees minus theta,

00:01:13.880 --> 00:01:16.810
you're going to get 90 degrees.

00:01:16.810 --> 00:01:18.900
Now, why is this interesting?

00:01:18.900 --> 00:01:26.520
Well, let's think about what
the sine of theta is equal to.

00:01:26.520 --> 00:01:29.730
Sine is opposite
over hypotenuse.

00:01:29.730 --> 00:01:33.160
The opposite side is BC.

00:01:33.160 --> 00:01:34.920
So this is going
to be the length

00:01:34.920 --> 00:01:38.830
of BC over the hypotenuse.

00:01:38.830 --> 00:01:41.740
The hypotenuse is side AB.

00:01:41.740 --> 00:01:46.040
So the length of BC
over the length of AB.

00:01:49.100 --> 00:01:50.830
Now, what is that
ratio if we were

00:01:50.830 --> 00:01:52.810
to look at this angle
right over here?

00:01:52.810 --> 00:01:56.560
Well, for angle B, BC
is the adjacent side,

00:01:56.560 --> 00:01:58.970
and AB is the hypotenuse.

00:01:58.970 --> 00:02:01.810
From angle B's
perspective, this is

00:02:01.810 --> 00:02:03.880
the adjacent over
the hypotenuse.

00:02:03.880 --> 00:02:07.777
Now, what trig ratio is
adjacent over hypotenuse?

00:02:07.777 --> 00:02:08.610
Well, that's cosine.

00:02:08.610 --> 00:02:11.500
Sohcahtoa, let me
write that down.

00:02:11.500 --> 00:02:12.130
Doesn't hurt.

00:02:12.130 --> 00:02:14.030
Sine is opposite
over hypotenuse.

00:02:14.030 --> 00:02:15.870
We see that right over there.

00:02:15.870 --> 00:02:18.940
Cosine is adjacent
over hypotenuse, cah.

00:02:18.940 --> 00:02:21.790
And toa, tangent is
opposite over adjacent.

00:02:21.790 --> 00:02:23.970
So from this
angle's perspective,

00:02:23.970 --> 00:02:28.240
taking the length of BC,
BC is its adjacent side,

00:02:28.240 --> 00:02:30.400
and the hypotenuse is still AB.

00:02:30.400 --> 00:02:32.030
So from this
angle's perspective,

00:02:32.030 --> 00:02:34.120
this is adjacent
over hypotenuse.

00:02:34.120 --> 00:02:35.620
Or another way of
thinking about it,

00:02:35.620 --> 00:02:38.100
it's the cosine of this angle.

00:02:38.100 --> 00:02:43.340
So that's going to be
equal to the cosine

00:02:43.340 --> 00:02:47.670
of 90 degrees minus theta.

00:02:47.670 --> 00:02:49.120
That's a pretty
neat relationship.

00:02:49.120 --> 00:02:54.620
The sine of an angle is equal
to the cosine of its complement.

00:02:54.620 --> 00:02:57.720
So one way to think about
it, the sine of-- we

00:02:57.720 --> 00:02:59.740
could just pick any
arbitrary angle-- let's say,

00:02:59.740 --> 00:03:03.530
the sine of 60
degrees is going to be

00:03:03.530 --> 00:03:04.984
equal to the cosine of what?

00:03:04.984 --> 00:03:07.400
And I encourage you to pause
the video and think about it.

00:03:07.400 --> 00:03:10.190
Well, it's going to be
the cosine of 90 minus 60.

00:03:10.190 --> 00:03:12.800
It's going to be the
cosine of 30 degrees.

00:03:12.800 --> 00:03:14.829
30 plus 60 is 90.

00:03:14.829 --> 00:03:16.870
And of course, you could
go the other way around.

00:03:16.870 --> 00:03:20.230
We could think about
the cosine of theta.

00:03:20.230 --> 00:03:25.170
The cosine of theta is going to
be equal to the adjacent side

00:03:25.170 --> 00:03:28.630
to theta, to angle
A, I should say.

00:03:28.630 --> 00:03:31.160
And so the adjacent
side is right over here.

00:03:31.160 --> 00:03:32.790
That's AC.

00:03:32.790 --> 00:03:36.420
So it's going to be AC
over the hypotenuse,

00:03:36.420 --> 00:03:37.910
adjacent over hypotenuse.

00:03:37.910 --> 00:03:41.110
The hypotenuse is AB.

00:03:41.110 --> 00:03:43.850
But what is this ratio from
angle B's point of view?

00:03:43.850 --> 00:03:47.920
Well, the sine of
angle B is going

00:03:47.920 --> 00:03:53.200
to be its opposite side,
AC, over the hypotenuse, AB.

00:03:53.200 --> 00:03:56.210
So this right over here,
from angle B's perspective,

00:03:56.210 --> 00:03:58.270
this is angle B's sine.

00:03:58.270 --> 00:04:07.700
So this is equal to the sine
of 90 degrees minus theta.

00:04:07.700 --> 00:04:09.760
So the cosine of an angle
is equal to the sine

00:04:09.760 --> 00:04:10.510
of its complement.

00:04:10.510 --> 00:04:13.873
The sine of an angle is equal
to the cosine of its complement.


Sine and cosine of complements example

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.990
We are told that the
cosine of 58 degrees

00:00:03.990 --> 00:00:07.037
is roughly equal to 0.53.

00:00:07.037 --> 00:00:08.870
And that's roughly equal
to, because it just

00:00:08.870 --> 00:00:10.140
keeps going on and on.

00:00:10.140 --> 00:00:13.250
I just rounded it to
the nearest hundredth.

00:00:13.250 --> 00:00:16.466
And then we're asked, what
is the sine of 32 degrees?

00:00:16.466 --> 00:00:18.090
And I encourage you
to pause this video

00:00:18.090 --> 00:00:19.260
and try it on your own.

00:00:19.260 --> 00:00:21.670
And a hint is to look
at this right triangle.

00:00:21.670 --> 00:00:24.356
One of the angles is
already labeled 32 degrees.

00:00:24.356 --> 00:00:25.730
Figure out what
all of the angles

00:00:25.730 --> 00:00:28.410
are, and then use the
fundamental definitions,

00:00:28.410 --> 00:00:30.890
your sohcahtoa definitions,
to see if you can figure out

00:00:30.890 --> 00:00:33.930
what sine of 32 degrees is.

00:00:33.930 --> 00:00:36.840
So I'm assuming you've
given a go at it.

00:00:36.840 --> 00:00:38.630
Let's work it through now.

00:00:38.630 --> 00:00:41.350
So we know that the sum of
the angles of a triangle

00:00:41.350 --> 00:00:42.780
add up to 180.

00:00:42.780 --> 00:00:46.000
Now in a right angle, one
of the angles is 90 degrees.

00:00:46.000 --> 00:00:50.650
So that means that the
other two must add up to 90.

00:00:50.650 --> 00:00:52.450
These two add up to
90 plus another 90

00:00:52.450 --> 00:00:54.130
is going to be 180 degrees.

00:00:54.130 --> 00:00:56.460
Or another way to think
about is that the other two

00:00:56.460 --> 00:00:59.920
non-right angles are
going to be complementary.

00:00:59.920 --> 00:01:03.724
So what plus 32 is equal to 90?

00:01:03.724 --> 00:01:06.490
Well, 90 minus 32 is 58.

00:01:06.490 --> 00:01:08.800
So this right over here
is going to be 58 degrees.

00:01:08.800 --> 00:01:10.050
Well, why is that interesting?

00:01:10.050 --> 00:01:12.870
Well, we already know what
the cosine of 58 degrees

00:01:12.870 --> 00:01:13.966
is equal to.

00:01:13.966 --> 00:01:15.340
But let's think
about it in terms

00:01:15.340 --> 00:01:19.280
of ratios of the lengths of
sides of this right triangle.

00:01:19.280 --> 00:01:20.680
Let's just write down sohcahtoa.

00:01:20.680 --> 00:01:22.860
Soh, sine, is opposite
over hypotenuse.

00:01:22.860 --> 00:01:25.750
Cah, cosine, is adjacent
over hypotenuse.

00:01:25.750 --> 00:01:28.940
Toa, tangent, is
opposite over adjacent.

00:01:28.940 --> 00:01:30.790
So we could write
down the cosine

00:01:30.790 --> 00:01:33.080
of 58 degrees, which
we already know.

00:01:33.080 --> 00:01:35.730
If we think about it in terms
of these fundamental ratios,

00:01:35.730 --> 00:01:37.930
cosine is adjacent
over hypotenuse.

00:01:37.930 --> 00:01:39.830
This is a 58 degree angle.

00:01:39.830 --> 00:01:42.600
The side that is adjacent
to it is-- let me do it

00:01:42.600 --> 00:01:48.177
in this color-- is side
BC right over here.

00:01:48.177 --> 00:01:50.010
It's one of the sides
of the angle, the side

00:01:50.010 --> 00:01:51.676
of the angle that is
not the hypotenuse.

00:01:51.676 --> 00:01:54.002
The other side, this over
here, is a hypotenuse.

00:01:54.002 --> 00:01:55.960
So this is going to be
the adjacent, the length

00:01:55.960 --> 00:02:00.440
of the adjacent side, BC, over
the length of the hypotenuse.

00:02:04.120 --> 00:02:09.139
The length of the
hypotenuse, well, that is AB.

00:02:09.139 --> 00:02:11.830
Now let's think about what the
sine of 32 degrees would be.

00:02:17.080 --> 00:02:19.740
Well, sine is opposite
over hypotenuse.

00:02:19.740 --> 00:02:21.710
So now we're looking at
this 32 degree angle.

00:02:21.710 --> 00:02:23.420
What side is opposite it?

00:02:23.420 --> 00:02:25.245
Well, it opens up onto BC.

00:02:30.520 --> 00:02:33.220
And what's the length
of the hypotenuse?

00:02:33.220 --> 00:02:35.970
It's AB.

00:02:35.970 --> 00:02:39.970
Notice, the sine of 32
degrees is BC over AB.

00:02:39.970 --> 00:02:43.750
The cosine of 58
degrees is BC over AB.

00:02:43.750 --> 00:02:46.740
Or another way of thinking
about it, the sine of this angle

00:02:46.740 --> 00:02:50.050
is the same thing as the
cosine of this angle.

00:02:50.050 --> 00:02:52.470
So we could literally
write the sine--

00:02:52.470 --> 00:03:01.310
I want to do that in that pink
color-- the sine of 32 degrees

00:03:01.310 --> 00:03:07.660
is equal to the cosine
of 58 degrees, which

00:03:07.660 --> 00:03:11.810
is roughly equal to 0.53.

00:03:11.810 --> 00:03:15.350
And this is a really,
really useful property.

00:03:15.350 --> 00:03:20.550
The sine of an angle is equal
to the cosine of its complement.

00:03:20.550 --> 00:03:23.080
So we could write
this in general terms.

00:03:23.080 --> 00:03:27.120
We could write that
the sine of some angle

00:03:27.120 --> 00:03:29.670
is equal to the cosine
of its complement,

00:03:29.670 --> 00:03:33.690
is equal to the cosine
of 90 minus theta.

00:03:33.690 --> 00:03:35.330
Think about it.

00:03:35.330 --> 00:03:38.670
I could change this
entire problem.

00:03:38.670 --> 00:03:41.110
Instead of making this
the sine of 32 degrees,

00:03:41.110 --> 00:03:44.300
I could make this the
sine of 25 degrees.

00:03:44.300 --> 00:03:48.130
And if someone gave you the
cosine of-- what's 90 minus

00:03:48.130 --> 00:03:52.080
25?-- if someone gave you
the cosine of 65 degrees,

00:03:52.080 --> 00:03:55.630
then you could think
about this as 25.

00:03:55.630 --> 00:03:58.090
The complement is going
to be right over here.

00:03:58.090 --> 00:04:00.410
This would be 65 degrees.

00:04:00.410 --> 00:04:04.080
And then you could use
the exact same idea.


Thiago asks: How much time does a goalkeeper have to react to a penalty kick?

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:08.480
THIAGO SILVA: [SPEAKING
FOREIGN LANGUAGE]

00:00:08.480 --> 00:00:10.260
SALMAN KHAN: Great
question, Tiago.

00:00:10.260 --> 00:00:13.150
And to understand it, let's look
at the dimensions of a penalty

00:00:13.150 --> 00:00:13.870
kick.

00:00:13.870 --> 00:00:18.740
The kick itself is from 12 yards
away from the goal or 36 feet.

00:00:18.740 --> 00:00:22.480
The goal itself is 24
feet wide or 8 yards wide.

00:00:22.480 --> 00:00:24.670
And the goal is 8 feet tall.

00:00:24.670 --> 00:00:27.070
And so let's think about a
few other dimensions that

00:00:27.070 --> 00:00:28.500
might not be as obvious.

00:00:28.500 --> 00:00:30.210
Let's try to figure
out the distance

00:00:30.210 --> 00:00:34.160
from the ball where it's
kicked to the bottom right

00:00:34.160 --> 00:00:34.722
of the goal.

00:00:34.722 --> 00:00:36.930
And this, obviously, is
going to be the same distance

00:00:36.930 --> 00:00:40.200
as to the bottom
left of the goal.

00:00:40.200 --> 00:00:42.990
And I encourage anyone watching
this to pause the video

00:00:42.990 --> 00:00:45.590
and think about that right now.

00:00:45.590 --> 00:00:47.190
Well, the way I've
drawn it, you see

00:00:47.190 --> 00:00:50.870
that this is actually
a right triangle.

00:00:50.870 --> 00:00:53.430
And so we could use
the Pythagorean theorem

00:00:53.430 --> 00:00:57.571
in order to figure out what is
this distance right over here.

00:00:57.571 --> 00:00:58.820
And you might say, well, wait.

00:00:58.820 --> 00:00:59.903
How do we figure that out?

00:00:59.903 --> 00:01:02.880
Well, we already know that
this length of the triangle

00:01:02.880 --> 00:01:04.510
is 36 feet.

00:01:04.510 --> 00:01:06.570
And we know that this
base right over here

00:01:06.570 --> 00:01:08.710
is 1/2 of the width of the goal.

00:01:08.710 --> 00:01:11.570
So this is going to be
12 feet right over there.

00:01:11.570 --> 00:01:13.250
So the Pythagorean
theorem tells us

00:01:13.250 --> 00:01:15.210
that this distance
right over here

00:01:15.210 --> 00:01:18.870
is going to be the square
root of the square of the sum

00:01:18.870 --> 00:01:20.550
of the squares of
the other side.

00:01:20.550 --> 00:01:22.550
So it's going to be
the square root of 12

00:01:22.550 --> 00:01:25.770
squared plus 36 squared.

00:01:25.770 --> 00:01:28.120
And let's get our calculator
out and try to answer it

00:01:28.120 --> 00:01:30.990
and figure out what that is.

00:01:30.990 --> 00:01:39.270
So that's going to be the square
root of 12 squared plus 36

00:01:39.270 --> 00:01:49.076
feet squared-- oops, not 33-- 36
feet squared is equal to 37.9.

00:01:49.076 --> 00:01:51.700
So let's just say-- well, let's
just use that number right now.

00:01:51.700 --> 00:01:56.690
So 37 point-- we could even
say 0.95, almost 38 feet.

00:01:56.690 --> 00:02:04.860
So this is approximately
equal to 37.95 feet.

00:02:04.860 --> 00:02:06.860
And that's going to be
the same as this distance

00:02:06.860 --> 00:02:08.119
right over here.

00:02:08.119 --> 00:02:09.910
Now, let's figure out
the distance, an even

00:02:09.910 --> 00:02:11.944
further distance, the
distance to the top right,

00:02:11.944 --> 00:02:13.610
which is also going
to be the same thing

00:02:13.610 --> 00:02:15.530
as the distance to the top left.

00:02:15.530 --> 00:02:18.110
And I'd encourage, once again,
people to pause the video

00:02:18.110 --> 00:02:21.180
and try to think about
that on their own.

00:02:21.180 --> 00:02:23.140
Well, let's draw
another right triangle.

00:02:23.140 --> 00:02:25.840
And this one might
not be as obvious.

00:02:25.840 --> 00:02:29.100
But if I draw the straight
line distance of the ball

00:02:29.100 --> 00:02:31.990
to, in this case, the
top right of the goal,

00:02:31.990 --> 00:02:34.410
I have now constructed
another right triangle.

00:02:34.410 --> 00:02:36.780
Notice this is a
90 degree angle.

00:02:36.780 --> 00:02:40.320
One side is 37.95 feet.

00:02:40.320 --> 00:02:42.580
The other side is
eight feet tall.

00:02:42.580 --> 00:02:46.260
And so this distance
right over here

00:02:46.260 --> 00:02:52.130
is going to be the
square root of 37.95

00:02:52.130 --> 00:02:57.975
squared plus 8 feet squared.

00:03:00.620 --> 00:03:04.600
So let's figure out what that
is, get the calculator out.

00:03:04.600 --> 00:03:07.180
So I can square the last
entry on my calculator

00:03:07.180 --> 00:03:09.360
just by typing in
this-- that just means

00:03:09.360 --> 00:03:11.810
take the last answer,
square it-- and then

00:03:11.810 --> 00:03:16.200
add that to 8 squared,
which we know is 64.

00:03:16.200 --> 00:03:18.290
And now we want to take
the square root of that.

00:03:18.290 --> 00:03:22.150
So take the square
root of 1,504,

00:03:22.150 --> 00:03:27.640
gets us to 38, we'll just
say, roughly 38.8 feet,

00:03:27.640 --> 00:03:29.750
or let's say 0.78 feet.

00:03:29.750 --> 00:03:35.590
So this is approximately
equal to 38.78 feet.

00:03:35.590 --> 00:03:37.340
Now the next thing I
want to think about--

00:03:37.340 --> 00:03:39.910
and I think this will be what
we focus on-- to figure out

00:03:39.910 --> 00:03:42.560
how much time does the goalie
have to get here, because one

00:03:42.560 --> 00:03:44.720
could argue that this is
the hardest to get to,

00:03:44.720 --> 00:03:47.900
that the goalie, they have
to travel the furthest.

00:03:47.900 --> 00:03:51.390
And they have to dive
for this right over here.

00:03:51.390 --> 00:03:53.270
So let's think
about the distance

00:03:53.270 --> 00:03:56.247
from this point to
this point here.

00:03:56.247 --> 00:03:58.580
And then we can think about
how much the goalie actually

00:03:58.580 --> 00:04:00.330
has to move because
they have some height.

00:04:00.330 --> 00:04:02.572
And their hands, they
can stick up in the air.

00:04:02.572 --> 00:04:04.780
And this one, once again,
is a fairly straightforward

00:04:04.780 --> 00:04:06.220
Pythagorean theorem problem.

00:04:06.220 --> 00:04:07.990
You have a right triangle here.

00:04:07.990 --> 00:04:09.840
You could also see
it on this side.

00:04:09.840 --> 00:04:11.280
It's a little easier.

00:04:11.280 --> 00:04:14.220
You have a right triangle here.

00:04:14.220 --> 00:04:16.089
We know that this is 12 feet.

00:04:16.089 --> 00:04:19.230
And that this right
over here is 8 feet.

00:04:19.230 --> 00:04:21.760
So we know that this
distance right over here

00:04:21.760 --> 00:04:25.440
is going to be the square
root of 12 feet squared,

00:04:25.440 --> 00:04:30.080
which is 144, plus 8 feet
squared, which is 64.

00:04:30.080 --> 00:04:32.100
So let's figure
out what that is.

00:04:32.100 --> 00:04:40.720
So that's going to be
the square root of 144

00:04:40.720 --> 00:04:48.430
plus 64 is equal to 14.42 feet.

00:04:48.430 --> 00:04:52.920
It's equal to 14.42 feet.

00:04:52.920 --> 00:04:56.760
Now, we assume that the goalie
isn't traveling all the way

00:04:56.760 --> 00:04:58.476
from here to all the way there.

00:04:58.476 --> 00:04:59.600
The goalie has some height.

00:04:59.600 --> 00:05:02.490
And he or she could stick
their hands up in the air.

00:05:02.490 --> 00:05:05.460
So we could imagine a goalie
stretched out like this,

00:05:05.460 --> 00:05:07.510
trying to dive for that ball.

00:05:07.510 --> 00:05:10.520
And so the actual distance
that they have to travel

00:05:10.520 --> 00:05:15.360
is from the tip of their
reach to that corner right

00:05:15.360 --> 00:05:16.360
over there.

00:05:16.360 --> 00:05:21.830
So if we assume that the entire,
if a goalie stretched out, is,

00:05:21.830 --> 00:05:25.690
let's say, 7 and 1/2 feet
completely stretched out--

00:05:25.690 --> 00:05:29.050
so this distance, if this
distance fully stretched

00:05:29.050 --> 00:05:37.140
out is 7.5 feet and they're
trying to get 14.42 feet away--

00:05:37.140 --> 00:05:38.920
and I could maybe
start rounding down

00:05:38.920 --> 00:05:42.560
a little bit rougher numbers
to, say, 14.4 feet-- then

00:05:42.560 --> 00:05:47.020
they need to travel
about 6.9 feet.

00:05:47.020 --> 00:05:51.450
So they need to
travel about 6.9 feet.

00:05:51.450 --> 00:05:54.580
So for this top right kick
or this top left kick,

00:05:54.580 --> 00:06:02.370
the ball is going to travel
38, almost 39 feet, 38.8,

00:06:02.370 --> 00:06:04.080
38.78 feet.

00:06:04.080 --> 00:06:07.080
And the goalie has to
travel-- the goaltender has

00:06:07.080 --> 00:06:12.230
to travel 6.9 feet.

00:06:12.230 --> 00:06:15.380
Now that we know the distances
that the ball has to travel

00:06:15.380 --> 00:06:17.610
and that the goalkeeper
has to travel,

00:06:17.610 --> 00:06:20.240
we can now think about the time
in which it's going to happen.

00:06:20.240 --> 00:06:22.530
And to do that, we're going
to have make assumptions

00:06:22.530 --> 00:06:24.060
about their speed.

00:06:24.060 --> 00:06:26.860
So I did some research
on the internet.

00:06:26.860 --> 00:06:31.660
And it looks like a penalty
kick can go-- a fast penalty

00:06:31.660 --> 00:06:34.860
kick can be around
60 miles per hour,

00:06:34.860 --> 00:06:37.800
although it does look like
there are documented cases of 80

00:06:37.800 --> 00:06:39.920
miles per hour or
even higher than that.

00:06:39.920 --> 00:06:44.000
But let's just say 60 miles per
hour for a fast penalty kick.

00:06:44.000 --> 00:06:46.460
So this is the kick
speed or the ball speed.

00:06:49.610 --> 00:06:51.560
And let's assume
that this person

00:06:51.560 --> 00:06:54.480
can jump at 15 miles per
hour, which is actually

00:06:54.480 --> 00:06:56.930
a pretty good speed
from a standstill.

00:06:56.930 --> 00:06:59.170
So it actually might be
a little bit aggressive.

00:06:59.170 --> 00:07:04.320
So jump speed--
I'll write it here--

00:07:04.320 --> 00:07:06.620
jump speed of the goalkeeper.

00:07:06.620 --> 00:07:09.396
Let's write that as
15 miles per hour.

00:07:09.396 --> 00:07:11.770
And so to make sense of it,
because everything else we've

00:07:11.770 --> 00:07:16.010
done in feet, let's convert
these each into feet.

00:07:16.010 --> 00:07:17.924
So 60 miles per hour.

00:07:17.924 --> 00:07:19.590
If I want to convert
it to feet, we just

00:07:19.590 --> 00:07:25.290
have to remind ourselves
that 60 miles is

00:07:25.290 --> 00:07:34.480
the equivalent to
60 times 5,280 feet.

00:07:34.480 --> 00:07:37.330
Each mile is 5,280 feet.

00:07:37.330 --> 00:07:40.050
So this would give me the total
number of feet in an hour.

00:07:40.050 --> 00:07:41.480
But we don't want feet per hour.

00:07:41.480 --> 00:07:43.710
We want feet per seconds.

00:07:43.710 --> 00:07:48.060
So this is how far you
would go in feet in an hour.

00:07:48.060 --> 00:07:50.320
To figure it out in seconds,
you would want to divide

00:07:50.320 --> 00:07:56.420
by 3,600 because there are
3,600 seconds in an hour.

00:07:56.420 --> 00:08:03.830
So this gets us to 88
feet per second, 88

00:08:03.830 --> 00:08:07.070
feet per second for the ball.

00:08:07.070 --> 00:08:09.980
And now let's do the same
thing for the goalkeeper.

00:08:09.980 --> 00:08:19.710
So 15 times 5,280-- so this is
the feet traveled in an hour.

00:08:19.710 --> 00:08:21.410
But we want it in a second.

00:08:21.410 --> 00:08:24.530
So we're going to divide
it by 3,600, gets us

00:08:24.530 --> 00:08:27.620
to 22 feet per second.

00:08:27.620 --> 00:08:36.059
So this is equal to 22 feet in
a second, 22 feet per second.

00:08:36.059 --> 00:08:38.789
So now we can use these speeds
to figure out how long will it

00:08:38.789 --> 00:08:41.470
take the ball to
go from this point

00:08:41.470 --> 00:08:43.752
all the way to the
top right corner.

00:08:43.752 --> 00:08:45.460
Well, we'll just have
to remind ourselves

00:08:45.460 --> 00:08:49.340
that distance is equal
to speed times time.

00:08:49.340 --> 00:08:50.840
Or if we want the
time, we just have

00:08:50.840 --> 00:08:53.800
to take the distance
an divide it by speed.

00:08:53.800 --> 00:09:01.750
So the time for the
ball-- so ball time-- is

00:09:01.750 --> 00:09:03.960
going to be equal to
38 point-- let's just

00:09:03.960 --> 00:09:09.410
go with 38.8 feet-- had to
make a lot of rough assumptions

00:09:09.410 --> 00:09:13.710
here anyway-- is going to be
equal to 38.8 feet divided

00:09:13.710 --> 00:09:25.030
by 88 feet per second,
which is equal to-- so 38.8

00:09:25.030 --> 00:09:30.847
divided by 88 gets
us 0.44 seconds.

00:09:30.847 --> 00:09:31.680
So let's write that.

00:09:31.680 --> 00:09:37.790
So that's 0.44 seconds
or 44/100 of a second,

00:09:37.790 --> 00:09:40.586
a little under half of a second
for this ball to reach there.

00:09:40.586 --> 00:09:42.460
Obviously, if the ball
was going even faster,

00:09:42.460 --> 00:09:43.668
it would take even less time.

00:09:43.668 --> 00:09:47.100
If it was going slower, it
would take a little more time.

00:09:47.100 --> 00:09:49.680
Now let's think about how far
it would take for this person

00:09:49.680 --> 00:09:52.430
to travel the 6.9 feet.

00:09:52.430 --> 00:10:02.210
So the goalkeeper time, goalie
time is equal to the 6.9 feet.

00:10:02.210 --> 00:10:03.770
We're assuming he's
kind of already

00:10:03.770 --> 00:10:05.580
in this position, kind
of already starting

00:10:05.580 --> 00:10:06.460
to stretch out.

00:10:06.460 --> 00:10:09.550
Or he stretches out while
he's in the air when

00:10:09.550 --> 00:10:10.512
he launches himself.

00:10:10.512 --> 00:10:12.220
So it's going to be--
and, obviously, I'm

00:10:12.220 --> 00:10:15.750
making a lot of rough
assumptions here-- 6.9 feet

00:10:15.750 --> 00:10:17.660
divided by 22 feet per second.

00:10:21.270 --> 00:10:29.670
So that gets us 6.9 divided
by 22 is equal to 0.31--

00:10:29.670 --> 00:10:36.540
I'll just round there--
is equal to 0.31 seconds.

00:10:36.540 --> 00:10:38.490
So just based on what
we saw, the ball's

00:10:38.490 --> 00:10:41.890
going to take 44/100 of
a second to get there.

00:10:41.890 --> 00:10:44.680
The goalkeeper, if we assume
the 15 miles per hour,

00:10:44.680 --> 00:10:47.860
is going to take 31/100
of a second to get there.

00:10:47.860 --> 00:10:52.330
And so they only
have the difference

00:10:52.330 --> 00:10:54.230
to make the decision
where to jump

00:10:54.230 --> 00:10:55.910
and, even frankly,
to start their jump,

00:10:55.910 --> 00:10:57.390
getting into the
jumping position,

00:10:57.390 --> 00:11:00.050
to kind of scrunch up
and jump a little bit.

00:11:00.050 --> 00:11:01.810
So the difference
between these two things

00:11:01.810 --> 00:11:07.040
is only-- let me write this in
a new color-- this is 13/100

00:11:07.040 --> 00:11:11.750
of a second to
make this decision.

00:11:11.750 --> 00:11:14.440
And that's why, frankly,
penalty kicks are successful

00:11:14.440 --> 00:11:15.870
so frequently.

00:11:15.870 --> 00:11:19.510
Most people's reaction time--
and even professional athletes

00:11:19.510 --> 00:11:21.250
does not-- professional
athletes get

00:11:21.250 --> 00:11:22.982
close to this in terms
of reaction time.

00:11:22.982 --> 00:11:24.940
I did a little bit of
research on the internet.

00:11:24.940 --> 00:11:28.730
Most other people's reaction
time is nowhere near this low.

00:11:28.730 --> 00:11:31.270
It's often double
this or higher.

00:11:31.270 --> 00:11:33.630
So even if they make
the exact right decision

00:11:33.630 --> 00:11:36.250
and even if they're able
to launch themselves up

00:11:36.250 --> 00:11:40.740
at 15 miles per hour, they have
little over a tenth of a second

00:11:40.740 --> 00:11:42.060
to make that decision.

00:11:42.060 --> 00:11:43.580
Now once again, I
want to emphasize,

00:11:43.580 --> 00:11:46.160
this was given all of these
assumptions that I made.

00:11:46.160 --> 00:11:48.460
You might want to lower
or higher this assumption

00:11:48.460 --> 00:11:52.127
of how fast they can jump.

00:11:52.127 --> 00:11:53.710
You might want to
increase or decrease

00:11:53.710 --> 00:11:56.240
the assumption about how
fast the ball is going.

00:11:56.240 --> 00:11:58.410
And you could also think
about different points

00:11:58.410 --> 00:12:01.990
on the goal to see which one,
based on your assumptions,

00:12:01.990 --> 00:12:05.120
might require different
reaction times.


Angle to aim to get alien

WEBVTT
Kind: captions
Language: en

00:00:01.140 --> 00:00:03.450
A tiny but horrible
alien is standing

00:00:03.450 --> 00:00:05.355
at the top of the
Eiffel Tower-- so this

00:00:05.355 --> 00:00:07.840
is where the tiny but
horrible alien is-- which

00:00:07.840 --> 00:00:10.670
is 324 meters tall--
and they label that,

00:00:10.670 --> 00:00:12.830
the height of the Eiffel
Tower-- and threatening

00:00:12.830 --> 00:00:15.210
to destroy the city of Paris.

00:00:15.210 --> 00:00:18.442
A Men In Black-- or
a Men In Black agent.

00:00:18.442 --> 00:00:20.650
I was about to say maybe it
should be a man in black.

00:00:20.650 --> 00:00:23.640
A Men In Black agent is
standing at ground level,

00:00:23.640 --> 00:00:26.840
54 meters across
the Eiffel square.

00:00:26.840 --> 00:00:29.340
So 54 meters from,
I guess you could

00:00:29.340 --> 00:00:31.800
say the center of the
base of the Eiffel Tower,

00:00:31.800 --> 00:00:33.630
aiming his laser
gun at the alien.

00:00:33.630 --> 00:00:36.140
So this is him
aiming the laser gun.

00:00:36.140 --> 00:00:39.420
At what angle should the
agent shoot his laser gun?

00:00:39.420 --> 00:00:43.580
Round your answer, if necessary,
to two decimal places.

00:00:43.580 --> 00:00:47.380
So if we construct a right
triangle here, and we can.

00:00:47.380 --> 00:00:53.990
So the height of this right
triangle is 324 meters.

00:00:53.990 --> 00:00:56.605
This width right over
here is 54 meters.

00:00:56.605 --> 00:00:59.230
It is a right triangle.

00:00:59.230 --> 00:01:01.210
What they're really
asking us is what

00:01:01.210 --> 00:01:06.620
is this angle right over here.

00:01:06.620 --> 00:01:09.060
And they've given us two
pieces of information.

00:01:09.060 --> 00:01:12.500
They gave us the side that
is opposite the angle.

00:01:12.500 --> 00:01:16.880
And they've given us the side
that is adjacent to the angle.

00:01:16.880 --> 00:01:20.620
So what trig function deals
with opposite and adjacent?

00:01:20.620 --> 00:01:22.780
And to remind
ourselves, we can write,

00:01:22.780 --> 00:01:26.080
like I always like
to do, soh, cah, toa.

00:01:26.080 --> 00:01:27.910
And these are really
by definition.

00:01:27.910 --> 00:01:31.180
So you just have to know this,
and soh cah toa helps us.

00:01:31.180 --> 00:01:33.390
Sine is opposite
over hypotenuse.

00:01:33.390 --> 00:01:35.900
Cosine is adjacent
over hypotenuse.

00:01:35.900 --> 00:01:39.475
Tangent is opposite
over adjacent.

00:01:43.330 --> 00:01:47.760
We can write that
the tangent of theta

00:01:47.760 --> 00:01:50.340
is equal to the length
of the opposite side--

00:01:50.340 --> 00:02:00.600
324 meters-- over the length
of the adjacent side-- over 54

00:02:00.600 --> 00:02:01.760
meters.

00:02:01.760 --> 00:02:03.520
Now you might say,
well, OK, that's fine.

00:02:03.520 --> 00:02:08.830
What angle, when I take its
tangent, gives me 324/54?

00:02:08.830 --> 00:02:12.217
Well, for this, it will probably
be useful to use a calculator.

00:02:12.217 --> 00:02:13.800
And the way that
we'd use a calculator

00:02:13.800 --> 00:02:16.810
is we would use the
Inverse Tan Function.

00:02:16.810 --> 00:02:19.460
So we could rewrite
this as we're

00:02:19.460 --> 00:02:21.320
going to take the
inverse tangent--

00:02:21.320 --> 00:02:24.970
and sometimes it's written as
tangent with this negative 1

00:02:24.970 --> 00:02:25.760
superscript.

00:02:25.760 --> 00:02:28.980
So the inverse tangent
of tan of theta

00:02:28.980 --> 00:02:32.503
is going to be equal to the
inverse tangent of 324/54.

00:02:40.065 --> 00:02:42.190
And just to be clear, what
is this inverse tangent?

00:02:42.190 --> 00:02:45.410
This just literally
says, this will

00:02:45.410 --> 00:02:49.530
return what is the angle that,
when I take the tangent of it,

00:02:49.530 --> 00:02:52.990
gives me 324/54.

00:02:52.990 --> 00:02:57.450
This says, what is
the angle that, when

00:02:57.450 --> 00:03:01.560
I take the tangent of it,
gives me tangent of theta?

00:03:01.560 --> 00:03:05.780
So this right over here, this
just simplifies to theta.

00:03:05.780 --> 00:03:08.840
Theta is the angle that when
you get the tangent of it

00:03:08.840 --> 00:03:10.590
gets you tangent of theta.

00:03:10.590 --> 00:03:15.220
And so we get theta is equal
to inverse tangent of 324/54.

00:03:19.990 --> 00:03:23.060
Once again, this inverse tangent
thing you might find confusing.

00:03:23.060 --> 00:03:25.020
But all this is saying
is, over here, we're

00:03:25.020 --> 00:03:28.350
saying tangent of
some angle is 324/54.

00:03:28.350 --> 00:03:30.695
This is just saying
my angle is whatever

00:03:30.695 --> 00:03:34.300
angle I need so that when
I take the tangent of it,

00:03:34.300 --> 00:03:36.540
I get 324/54.

00:03:36.540 --> 00:03:38.770
It's how we will
solve for theta.

00:03:38.770 --> 00:03:40.310
So let's get our calculator out.

00:03:40.310 --> 00:03:45.950
And let's say that we want
our answer in degrees.

00:03:45.950 --> 00:03:48.020
Well, I'm just going
to assume that they

00:03:48.020 --> 00:03:49.310
want our answers in degrees.

00:03:49.310 --> 00:03:51.710
So let me make sure my
calculator is actually

00:03:51.710 --> 00:03:54.370
in degree mode.

00:03:54.370 --> 00:03:56.344
So I'll go to the 2nd
mode right over here.

00:03:56.344 --> 00:03:58.135
And actually it's in
radian mode right now.

00:03:58.135 --> 00:03:59.850
So let me make sure
I'm in degree mode

00:03:59.850 --> 00:04:01.635
to get my answer in degrees.

00:04:01.635 --> 00:04:03.310
Now let me exit out of here.

00:04:03.310 --> 00:04:05.942
And let me just type in
the inverse tangent--

00:04:05.942 --> 00:04:07.650
so it's in this yellow
color right here--

00:04:07.650 --> 00:04:13.360
inverse tangent of
324 divided by 54

00:04:13.360 --> 00:04:17.160
is going to be-- and
they told us to round

00:04:17.160 --> 00:04:21.680
to two decimal
places-- 80.54 degrees.

00:04:21.680 --> 00:04:26.640
So theta is equal
to 80.54 degrees.

00:04:26.640 --> 00:04:29.680
That's the angle at which
you should shoot the gun

00:04:29.680 --> 00:04:34.145
to help defeat this
horrible alien.


How much of a pyramid is submerged

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.120
The Nile River has
overflowed and covered

00:00:03.120 --> 00:00:05.380
its entire surroundings,
except for the tip

00:00:05.380 --> 00:00:07.780
of the Great Pyramid
in Giza, Egypt.

00:00:07.780 --> 00:00:12.180
An expedition was sent to find
how high the water had risen.

00:00:12.180 --> 00:00:14.590
The people measured the
edge of the pyramid that's

00:00:14.590 --> 00:00:17.890
above the water and found
it was 72 meters long.

00:00:17.890 --> 00:00:20.720
So this distance right
over here is 72 meters.

00:00:20.720 --> 00:00:22.610
They knew that the
entire length of the edge

00:00:22.610 --> 00:00:24.870
is 180 meters, when
it's not flooded.

00:00:24.870 --> 00:00:26.911
So this entire
length is 180 meters.

00:00:26.911 --> 00:00:29.160
They also knew that the
vertical height of the pyramid

00:00:29.160 --> 00:00:30.760
is 139 meters.

00:00:30.760 --> 00:00:32.800
So this is 139 meters.

00:00:32.800 --> 00:00:35.350
What is the level of the
water above the ground?

00:00:35.350 --> 00:00:38.006
So the ground is right over
here at the base of the pyramid.

00:00:38.006 --> 00:00:40.380
And so they want the level of
the water above the ground.

00:00:40.380 --> 00:00:42.910
So that's this height, this
height right over here.

00:00:42.910 --> 00:00:44.270
So let's just call that h.

00:00:44.270 --> 00:00:46.710
We want to figure out what h is.

00:00:46.710 --> 00:00:50.450
Round your answer, if necessary,
to two decimal places.

00:00:50.450 --> 00:00:52.420
So what do we know, and
what do we not know?

00:00:52.420 --> 00:00:55.230
So they've labeled this
little angle here theta.

00:00:55.230 --> 00:00:57.470
And this, of course,
is a right angle.

00:00:57.470 --> 00:00:59.757
So this angle here at
the base of the pyramid,

00:00:59.757 --> 00:01:01.590
this is going to be the
complement of theta.

00:01:01.590 --> 00:01:05.170
It's going to be 90
degrees minus theta.

00:01:05.170 --> 00:01:07.200
And using that
information, we can also

00:01:07.200 --> 00:01:11.570
figure out that this angle up
here is also going to be theta.

00:01:11.570 --> 00:01:15.170
If that looks a little
bit strange to you,

00:01:15.170 --> 00:01:19.640
let me just draw it out here and
make it a little bit clearer.

00:01:19.640 --> 00:01:21.820
If we have a triangle,
a right triangle,

00:01:21.820 --> 00:01:25.849
where this angle right over
here is 90 minus theta,

00:01:25.849 --> 00:01:27.890
and we wanted to figure
out what this is up here,

00:01:27.890 --> 00:01:29.400
let's say this is x.

00:01:29.400 --> 00:01:32.980
Well, we could say x
plus 90 minus theta,

00:01:32.980 --> 00:01:38.386
90 degrees minus
theta, plus 90 degrees

00:01:38.386 --> 00:01:40.010
is going to be equal
to-- well, the sum

00:01:40.010 --> 00:01:44.520
of the angles of a triangle
are going to be 180 degrees.

00:01:44.520 --> 00:01:48.060
Well, if we subtract 180 from
both sides, so that's 100,

00:01:48.060 --> 00:01:48.780
and that's 100.

00:01:48.780 --> 00:01:51.440
That's 180 from the
left, 180 from the right.

00:01:51.440 --> 00:01:55.640
We get x minus
theta is equal to 0.

00:01:55.640 --> 00:01:59.100
Or if you add theta to both
sides, x is equal to theta.

00:01:59.100 --> 00:02:02.250
So this thing up here is
going to be theta as well.

00:02:02.250 --> 00:02:04.550
So this is also
going to be theta.

00:02:04.550 --> 00:02:06.100
And what else do we know?

00:02:06.100 --> 00:02:07.630
Well, we know this is 72.

00:02:07.630 --> 00:02:09.949
We know that the
whole thing is 180.

00:02:09.949 --> 00:02:12.540
So this is 72, and the
whole thing is 180.

00:02:12.540 --> 00:02:16.140
The part of this edge that's
below the water, this distance

00:02:16.140 --> 00:02:16.807
right over here.

00:02:16.807 --> 00:02:18.848
Let me draw it without
cluttering the picture too

00:02:18.848 --> 00:02:19.575
much.

00:02:19.575 --> 00:02:21.020
I'll do it in that black color.

00:02:21.020 --> 00:02:25.130
This distance right over
here is going to be 108.

00:02:25.130 --> 00:02:28.140
108 plus 72 is 180.

00:02:28.140 --> 00:02:30.170
So what does this do for us?

00:02:30.170 --> 00:02:33.840
We need to figure
out this height.

00:02:33.840 --> 00:02:37.325
We know that this right over
here is a right triangle.

00:02:39.870 --> 00:02:42.340
I could color this in just to
make it a little bit clearer.

00:02:42.340 --> 00:02:46.190
This thing in yellow, right
over here is a right triangle.

00:02:46.190 --> 00:02:48.030
If we look at that
right triangle,

00:02:48.030 --> 00:02:52.710
and if we wanted to solve for
h and solve for h using a trig

00:02:52.710 --> 00:02:56.790
ratio based on this angle
theta right over here,

00:02:56.790 --> 00:03:02.270
we know that relative
to this angle theta,

00:03:02.270 --> 00:03:05.790
this side of length h
is an adjacent side.

00:03:05.790 --> 00:03:09.190
And this length of
108 right over here

00:03:09.190 --> 00:03:10.950
along the edge,
that's the hypotenuse

00:03:10.950 --> 00:03:13.650
of this yellow triangle
that I just highlighted in.

00:03:13.650 --> 00:03:18.390
So which trig ratio involves an
adjacent side and a hypotenuse?

00:03:18.390 --> 00:03:19.885
Well, we just write SOHCAHTOA.

00:03:23.280 --> 00:03:25.220
Sine is opposite
over hypotenuse.

00:03:25.220 --> 00:03:27.680
That would be this distance
over the hypotenuse.

00:03:27.680 --> 00:03:32.020
Cosine is adjacent
over hypotenuse.

00:03:32.020 --> 00:03:37.750
So we get the cosine of theta is
going to be equal to the height

00:03:37.750 --> 00:03:38.932
that we care about.

00:03:38.932 --> 00:03:40.890
That's the adjacent side
of this right triangle

00:03:40.890 --> 00:03:44.982
over the length of the
hypotenuse, OVER 108.

00:03:44.982 --> 00:03:46.690
Well, that doesn't
help us yet because we

00:03:46.690 --> 00:03:48.910
don't know what the
cosine of theta is.

00:03:48.910 --> 00:03:49.910
But there's a clue here.

00:03:49.910 --> 00:03:51.460
Theta is also sitting up here.

00:03:51.460 --> 00:03:53.760
So maybe if we can figure
out what cosine of theta

00:03:53.760 --> 00:03:56.310
is based up here, then
we can solve for h.

00:03:56.310 --> 00:03:58.944
So if we look at this data,
what is the cosine of theta?

00:03:58.944 --> 00:04:01.110
And now we're looking at a
different right triangle.

00:04:04.450 --> 00:04:07.300
We're looking at this
entire right triangle now.

00:04:07.300 --> 00:04:09.600
Based on that entire
right triangle,

00:04:09.600 --> 00:04:11.320
what is cosine of theta?

00:04:11.320 --> 00:04:13.150
Well, cosine of
theta, once again,

00:04:13.150 --> 00:04:15.360
is equal to adjacent
over hypotenuse.

00:04:15.360 --> 00:04:17.980
The adjacent length is this
length right over here.

00:04:17.980 --> 00:04:20.519
We already know
that's 139 meters.

00:04:20.519 --> 00:04:23.440
So it's going to be
equal to 139 meters.

00:04:23.440 --> 00:04:25.790
And what's the length
of the hypotenuse?

00:04:25.790 --> 00:04:28.280
Well, the hypotenuse is
this length right over here.

00:04:28.280 --> 00:04:31.990
It's 72 plus 108.

00:04:31.990 --> 00:04:33.490
Oh, we already have
it labeled here.

00:04:33.490 --> 00:04:34.340
It's 180.

00:04:34.340 --> 00:04:36.350
We can assume that
this is an isosceles--

00:04:36.350 --> 00:04:39.600
that this pyramid is
an isosceles triangle.

00:04:39.600 --> 00:04:42.300
So 180 on that side
and 180 on that side.

00:04:42.300 --> 00:04:44.850
So the cosine is
adjacent-- 139--

00:04:44.850 --> 00:04:48.400
over the hypotenuse,
which is 180, over 180.

00:04:48.400 --> 00:04:49.890
And these data
are the same data.

00:04:49.890 --> 00:04:51.210
We just showed that.

00:04:51.210 --> 00:04:54.560
So now we have cosine
of theta is h/108.

00:04:54.560 --> 00:04:56.890
Cosine of theta is 139/180.

00:04:56.890 --> 00:05:01.730
Or we could say that h/108,
which is cosine of theta,

00:05:01.730 --> 00:05:05.370
also is equal to 139/180.

00:05:05.370 --> 00:05:08.210
Both of these things are
equal to cosine of theta.

00:05:08.210 --> 00:05:11.160
Now to solve for h, we just
multiply both sides by 108.

00:05:11.160 --> 00:05:17.440
So h is equal to
139 times 108/180.

00:05:17.440 --> 00:05:20.150
So let's get our calculator
out and calculate that.

00:05:20.150 --> 00:05:30.060
So that is going to be 139
times 108, divided by 180,

00:05:30.060 --> 00:05:33.910
gets us to 83.4 meters.

00:05:33.910 --> 00:05:38.110
So h is equal to 83.4 meters.

00:05:38.110 --> 00:05:41.340
The height of the water is 83.4.


Example with trig functions and ratios

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.500
The diagram below contains
parallelogram ABCD,

00:00:04.500 --> 00:00:08.490
so that's parallelogram
ABCD, and triangle EFG.

00:00:08.490 --> 00:00:10.140
And they tell us
which of the angles

00:00:10.140 --> 00:00:12.348
have a measure of 90 degrees,
and they label it here,

00:00:12.348 --> 00:00:13.070
as well.

00:00:13.070 --> 00:00:14.200
And they tell us
which of the angles

00:00:14.200 --> 00:00:16.520
have a measure of 31 degrees,
and they label it here,

00:00:16.520 --> 00:00:17.690
as well.

00:00:17.690 --> 00:00:20.120
Which of the equalities
must be true?

00:00:20.120 --> 00:00:22.830
I encourage you to
now pause this video

00:00:22.830 --> 00:00:26.200
and try to figure
this out on your own.

00:00:26.200 --> 00:00:27.950
So now let's look at
this first statement,

00:00:27.950 --> 00:00:30.190
tangent of angle ADC.

00:00:30.190 --> 00:00:33.640
So let's think about the
tangent of angle ADC.

00:00:36.340 --> 00:00:40.170
So this is angle ADC
right here-- ADC.

00:00:40.170 --> 00:00:43.120
To remind ourselves the
definition of tangent,

00:00:43.120 --> 00:00:45.460
we'll break out sohcahtoa.

00:00:45.460 --> 00:00:47.830
Sine is opposite
over hypotenuse,

00:00:47.830 --> 00:00:52.770
cosine-- soh cah-- cosine
is adjacent over hypotenuse,

00:00:52.770 --> 00:00:56.490
toa-- tangent is
opposite over adjacent.

00:00:56.490 --> 00:00:58.920
So what is the opposite
side to this angle?

00:00:58.920 --> 00:01:02.700
So we're dealing with this
right triangle, triangle ADC.

00:01:02.700 --> 00:01:04.390
Let me highlight
it so that we know

00:01:04.390 --> 00:01:06.970
we're dealing with this right
triangle right over here.

00:01:06.970 --> 00:01:10.840
That's the only right triangle
that angle ADC is part of.

00:01:10.840 --> 00:01:15.510
And so what side is
opposite angle ADC?

00:01:15.510 --> 00:01:21.750
Well, it's side CA, or I
guess I say AC, side AC.

00:01:21.750 --> 00:01:23.590
So that is opposite.

00:01:23.590 --> 00:01:27.280
And what side is adjacent?

00:01:27.280 --> 00:01:30.516
Well, this side, CD.

00:01:30.516 --> 00:01:34.040
CD, or I guess I could call it
DC, whatever I want to call it.

00:01:34.040 --> 00:01:37.460
DC, or CD, is adjacent.

00:01:37.460 --> 00:01:39.880
Now how did I know that
this side is adjacent

00:01:39.880 --> 00:01:41.382
and not side DA?

00:01:41.382 --> 00:01:43.400
Because DA is the hypotenuse.

00:01:43.400 --> 00:01:46.640
They both, together, make up
the two sides of this angle.

00:01:46.640 --> 00:01:49.830
But the adjacent side
is one of the sides

00:01:49.830 --> 00:01:51.730
of the angle that is
not the hypotenuse.

00:01:51.730 --> 00:01:54.130
AD or DA in the
sohcahtoa context

00:01:54.130 --> 00:01:55.870
we would consider to
be the hypotenuse.

00:01:55.870 --> 00:01:58.040
For this angle, this is
opposite, this is adjacent,

00:01:58.040 --> 00:01:59.200
this is hypotenuse.

00:01:59.200 --> 00:02:03.530
Tangent of this angle is
opposite over adjacent-- AC

00:02:03.530 --> 00:02:05.385
over DC.

00:02:05.385 --> 00:02:06.760
Now is that what
they wrote here?

00:02:06.760 --> 00:02:07.260
No.

00:02:07.260 --> 00:02:09.470
They wrote AC over EF.

00:02:09.470 --> 00:02:10.199
Well, where's EF?

00:02:10.199 --> 00:02:12.990
EF is nowhere to be seen
either in this triangle,

00:02:12.990 --> 00:02:14.460
or even in this figure.

00:02:14.460 --> 00:02:17.830
EF is this thing
right over here.

00:02:17.830 --> 00:02:20.170
EF is this business
right over here.

00:02:20.170 --> 00:02:21.149
That's EF.

00:02:21.149 --> 00:02:22.773
It's in a completely
different triangle

00:02:22.773 --> 00:02:24.148
in a completely
different figure.

00:02:24.148 --> 00:02:26.240
We don't even know what
scale this is drawn at.

00:02:26.240 --> 00:02:28.030
There's no way the
tangent of this angle

00:02:28.030 --> 00:02:30.801
is related to this somewhat
arbitrary number that's

00:02:30.801 --> 00:02:31.300
over here.

00:02:31.300 --> 00:02:32.341
They haven't labelled it.

00:02:32.341 --> 00:02:35.650
This thing might be a million
miles long for all we know.

00:02:35.650 --> 00:02:39.470
This thing really
could be any number.

00:02:39.470 --> 00:02:42.040
So this isn't the case.

00:02:42.040 --> 00:02:44.740
We would have to relate it to
something within this triangle,

00:02:44.740 --> 00:02:46.360
or something that's
the same length.

00:02:46.360 --> 00:02:48.990
So if somehow we
could prove that EF

00:02:48.990 --> 00:02:52.249
is the same length as DC,
then we could go with that.

00:02:52.249 --> 00:02:53.040
But there's no way.

00:02:53.040 --> 00:02:55.581
This is a completely different
figure, a completely different

00:02:55.581 --> 00:02:56.470
diagram.

00:02:56.470 --> 00:02:58.170
This is a similar
triangle to this,

00:02:58.170 --> 00:02:59.700
but we don't know anything
about the lengths.

00:02:59.700 --> 00:03:02.200
A similar triangle just lets
us know that the angles are all

00:03:02.200 --> 00:03:04.790
the same, or that the ratio
between corresponding sides

00:03:04.790 --> 00:03:06.280
might be the same,
but it doesn't

00:03:06.280 --> 00:03:08.760
tell us what this number right
over here, doesn't tell us

00:03:08.760 --> 00:03:12.050
that this side is
somehow congruent to DC.

00:03:12.050 --> 00:03:14.190
So we can't go with this one.

00:03:14.190 --> 00:03:17.000
Now let's think about
the sine of CBA.

00:03:17.000 --> 00:03:19.500
So the sine-- let me do
this in a different color.

00:03:19.500 --> 00:03:20.980
So the sine of angle CBA.

00:03:26.860 --> 00:03:30.160
So that's this angle
right over here, CBA.

00:03:30.160 --> 00:03:32.670
Well, sine is opposite
over hypotenuse.

00:03:36.914 --> 00:03:38.330
I guess let me
make it clear which

00:03:38.330 --> 00:03:38.970
triangle we're looking at.

00:03:38.970 --> 00:03:40.100
I'll do this in yellow.

00:03:40.100 --> 00:03:43.300
We're now looking at this
triangle right over here.

00:03:43.300 --> 00:03:49.190
The opposite side is AC.

00:03:49.190 --> 00:03:51.480
That's what the
angle opens up into.

00:03:51.480 --> 00:03:54.230
So it's going to be equal to AC.

00:03:54.230 --> 00:03:57.140
And what is the hypotenuse?

00:03:57.140 --> 00:04:00.150
What is the hypotenuse here?

00:04:00.150 --> 00:04:02.060
Well, the hypotenuse--
so let me see,

00:04:02.060 --> 00:04:05.826
it's opposite over hypotenuse--
the hypotenuse is BC.

00:04:09.220 --> 00:04:12.160
It's the side opposite
the 90 degree side.

00:04:12.160 --> 00:04:15.770
So this, it's BC.

00:04:15.770 --> 00:04:24.490
Sine is opposite over
hypotenuse, so over BC.

00:04:24.490 --> 00:04:25.970
Is that what they
wrote over here?

00:04:25.970 --> 00:04:26.470
No.

00:04:26.470 --> 00:04:28.100
They have DC over BC.

00:04:28.100 --> 00:04:30.380
Now what is DC equal to?

00:04:30.380 --> 00:04:32.100
Well, DC is this.

00:04:32.100 --> 00:04:36.180
And DC is not-- there's no
evidence on this drawing

00:04:36.180 --> 00:04:40.880
right over here that DC is
somehow equivalent to AC.

00:04:40.880 --> 00:04:42.820
So given this information
right over here,

00:04:42.820 --> 00:04:44.970
we can't make this
statement, either.

00:04:44.970 --> 00:04:48.240
So neither of these are true.

00:04:48.240 --> 00:04:50.520
So let's make sure
we got this right.

00:04:50.520 --> 00:04:53.410
We can go back to
our actual exercise,

00:04:53.410 --> 00:04:58.040
and we get-- oh, that's
not the actual exercise.

00:04:58.040 --> 00:05:02.050
Let me minimize that.

00:05:02.050 --> 00:05:04.990
This is neither
of these are true.

00:05:04.990 --> 00:05:06.990
And we got it right.