Proof of angle addition formula for cosine

WEBVTT
Kind: captions
Language: en

00:00:00.706 --> 00:00:02.033
Voiceover: In the last video we proved

00:00:02.033 --> 00:00:04.435
the angle addition formula for sine.

00:00:04.435 --> 00:00:05.920
You could imagine in this video

00:00:05.920 --> 00:00:08.774
I would like to prove the angle addition for cosine,

00:00:08.774 --> 00:00:10.301
or in particular, that the cosine

00:00:10.301 --> 00:00:19.839
of X plus Y, of X plus Y, is equal to the cosine of X.

00:00:19.839 --> 00:00:25.510
Cosine of X, cosine of Y, cosine of Y minus,

00:00:25.510 --> 00:00:27.980
so if we have a plus here we're going to have a minus here,

00:00:27.980 --> 00:00:33.567
minus sine of X, sine of X, sine of Y.

00:00:33.567 --> 00:00:35.906
I'm going to use a very similar technique

00:00:35.906 --> 00:00:37.771
to the way I proved it for sine.

00:00:37.771 --> 00:00:39.044
I encourage you to pause the video

00:00:39.044 --> 00:00:41.847
either now or at any time that you get the inspiration

00:00:41.847 --> 00:00:45.205
to see if you can do this proof on your own.

00:00:45.577 --> 00:00:47.573
Just like we thought about it for sine,

00:00:47.573 --> 00:00:49.306
what is the cosine of X plus Y

00:00:49.306 --> 00:00:51.300
^in this diagram right over here?

00:00:51.300 --> 00:00:53.841
^Well, X plus Y is this angle right over here.

00:00:53.841 --> 00:00:59.712
^If we look at the right triangle ADF, cosine of X plus Y.

00:00:59.712 --> 00:01:04.907
^Well, cosine is adjacent over hypotenuse segment AF

00:01:04.907 --> 00:01:08.044
^over the hypotenuse, or since the hypotenuse

00:01:08.044 --> 00:01:12.443
^is just one, AF divided by one is just going to be AF.

00:01:12.443 --> 00:01:15.438
^Cosine of X plus Y is just the length of segment.

00:01:15.438 --> 00:01:18.510
^It's just the length of segment AF.

00:01:18.510 --> 00:01:22.439
^That right over there is equivalent to this right over here.

00:01:22.439 --> 00:01:26.638
Let me actually write that down; copy and paste.

00:01:26.638 --> 00:01:32.366
^Length of segment AF is cosine of X plus Y.

00:01:32.366 --> 00:01:34.706
^Let's think about how we can get that.

00:01:34.706 --> 00:01:36.700
^The way I'm going to think about it is,

00:01:36.700 --> 00:01:39.773
^given the other right triangles we have in this diagram,

00:01:39.773 --> 00:01:42.836
^if we could figure out that or AF.

00:01:42.836 --> 00:01:43.907
Let me write it this way.

00:01:43.907 --> 00:01:47.511
This thing, which is the same thing as AF, is equal to.

00:01:47.511 --> 00:01:50.708
Let me write it this way, it's equal to length of segment AB.

00:01:50.708 --> 00:01:54.846
It's equal to the length of segment AB,

00:01:54.846 --> 00:01:58.510
^which is this entire segment right over here,

00:01:58.510 --> 00:02:01.243
^minus the length of segment FB.

00:02:01.243 --> 00:02:05.843
^Minus this segment right over here.

00:02:05.843 --> 00:02:08.778
Minus the length of segment FB.

00:02:08.778 --> 00:02:11.911
Just from the way our angle addition formula

00:02:11.911 --> 00:02:13.567
looks for cosine, you might guess

00:02:13.567 --> 00:02:16.437
what's going to be AB and what's going to be FB.

00:02:16.437 --> 00:02:20.977
If we can prove that AB is equal to this.

00:02:20.977 --> 00:02:26.373
If we can prove that FB is equal to this, then we're done.

00:02:26.373 --> 00:02:28.245
Because we know that cosine of X plus Y,

00:02:28.245 --> 00:02:33.977
^which is AF, is equal to AB minus FB.

00:02:33.977 --> 00:02:38.641
If we can prove this, that it's equal to that minus that.

00:02:38.641 --> 00:02:41.505
Let's think about what these things actually are.

00:02:41.505 --> 00:02:43.441
What is AB?

00:02:43.441 --> 00:02:46.645
^Let's look at right triangle ACB.

00:02:46.645 --> 00:02:48.309
We know from the previous video that

00:02:48.309 --> 00:02:53.111
since triangle ADC has a hypotenuse of one,

00:02:53.111 --> 00:02:56.175
this length is one, that AC is cosine of X.

00:02:56.175 --> 00:03:00.244
^What is AB, well, think about it.

00:03:00.244 --> 00:03:05.040
^AB is adjacent to the angle that has measure Y.

00:03:05.040 --> 00:03:10.643
We could say that the cosine, let me do it down here.

00:03:10.643 --> 00:03:12.846
We could say, since I've already looked at all of that.

00:03:12.846 --> 00:03:16.241
^We could say that the cosine of Y,

00:03:16.241 --> 00:03:18.706
^cosine of Y is equal to it's adjacent side.

00:03:18.706 --> 00:03:20.645
^The length of it's adjacent side.

00:03:20.645 --> 00:03:25.113
^That is segment AB over the hypotenuse

00:03:25.113 --> 00:03:28.500
^over cosine of X, cosine of X.

00:03:28.500 --> 00:03:30.906
^Or multiply both sides by cosine of X.

00:03:30.906 --> 00:03:36.442
^We get that segment AB is equal to cosine of X,

00:03:36.442 --> 00:03:42.701
^cosine of X times cosine of Y, cosine of Y,

00:03:42.701 --> 00:03:45.305
^which is exactly what we set out to prove.

00:03:45.305 --> 00:03:49.034
We've just proven that AB is indeed

00:03:49.034 --> 00:03:51.112
the length of segment AB is indeed

00:03:51.112 --> 00:03:53.512
equal to cosine X, cosine Y.

00:03:53.512 --> 00:03:57.774
^This whole thing is equal to cosine X, cosine Y.

00:03:57.774 --> 00:03:59.580
^Now we just have to prove that segment

00:03:59.580 --> 00:04:01.574
FB is equal to sine X, sine Y.

00:04:01.574 --> 00:04:06.444
^This looks like a bit of a strange segment right over here.

00:04:06.444 --> 00:04:08.579
It's not part of any at least right triangle

00:04:08.579 --> 00:04:10.374
where we know that I've drawn

00:04:10.374 --> 00:04:12.170
where we know one of the angles.

00:04:12.170 --> 00:04:15.166
We can see from this diagram ECBF is a rectangle.

00:04:15.166 --> 00:04:18.036
We use that fact in the proof for

00:04:18.036 --> 00:04:19.707
the angle addition formula for sine.

00:04:19.707 --> 00:04:22.644
We'll also use it now because that tells us

00:04:22.644 --> 00:04:29.302
that FB is the same as EC, is EC.

00:04:29.302 --> 00:04:32.976
What is EC going to be equal to?

00:04:32.976 --> 00:04:36.512
Well, we have this angle Y right over here.

00:04:36.512 --> 00:04:42.303
What is, let's see, this side is opposite the angle Y.

00:04:42.303 --> 00:04:44.179
We might want to involve sine.

00:04:44.179 --> 00:04:48.695
We know that sine of Y, sine of Y,

00:04:48.695 --> 00:04:50.176
I'm looking right over here, is equal to

00:04:50.176 --> 00:04:53.571
the length of the opposite side, which is the length of EC,

00:04:53.571 --> 00:04:57.701
which is the length of EC over the hypotenuse,

00:04:57.701 --> 00:05:00.143
which is sine of X, sine of X.

00:05:00.143 --> 00:05:02.578
We figured that out from the last video.

00:05:02.578 --> 00:05:06.229
If this is X, opposite over hypotenuse is sine of X.

00:05:06.229 --> 00:05:07.579
Well, the opposite is just one,

00:05:07.579 --> 00:05:09.315
so the opposite is equal to sine of X.

00:05:09.407 --> 00:05:13.046
Over here multiply both sides by sine of X,

00:05:13.046 --> 00:05:14.577
and we get what we were looking for.

00:05:14.577 --> 00:05:20.512
EC is equal to sine of X, sine of X,

00:05:20.512 --> 00:05:25.513
times sine of Y, times sine of Y.

00:05:25.513 --> 00:05:27.575
Once again EC was the exact same thing.

00:05:27.575 --> 00:05:31.040
Has the same length as segment FB.

00:05:31.040 --> 00:05:33.397
We have just shown that segment FB

00:05:33.397 --> 00:05:35.434
is equal to sine of X times sine of Y.

00:05:35.434 --> 00:05:38.636
This is equal to that right over there.

00:05:38.636 --> 00:05:41.042
Once again, cosine of X plus Y,

00:05:41.042 --> 00:05:42.505
which is equal to segment AF,

00:05:42.505 --> 00:05:47.169
is equal to the length of segment AB

00:05:47.169 --> 00:05:50.577
minus the length of segment FB,

00:05:50.577 --> 00:05:52.507
which is equal to, we've proven the length

00:05:52.507 --> 00:05:54.705
of segment AB is cosine X, cosine Y,

00:05:54.705 --> 00:05:56.907
minus the length of segment FB,

00:05:56.907 --> 00:06:00.608
which is sine of X, sine of Y, and we are done.


Proof of angle addition formula for sine

WEBVTT
Kind: captions
Language: en

00:00:00.723 --> 00:00:02.831
Voiceover: What I hope to do
in this video is prove the

00:00:02.831 --> 00:00:05.554
angle addition formula for sine, or in

00:00:05.554 --> 00:00:10.522
particular prove that the sine of x plus y

00:00:15.160 --> 00:00:21.332
is equal to the sine of x times

00:00:21.582 --> 00:00:23.957
the cosine of -- I forgot my x.

00:00:23.957 --> 00:00:27.285
Sine of x times the cosine of y

00:00:29.331 --> 00:00:35.644
plus cosine of x times the sine of y.

00:00:38.471 --> 00:00:40.315
The way I'm going to do it is

00:00:40.315 --> 00:00:42.303
with this diagram right over here.

00:00:42.303 --> 00:00:46.613
You can view it as, it has
this red right triangle.

00:00:47.096 --> 00:00:50.301
It has this right triangle
that has a hypotenuse of one.

00:00:50.500 --> 00:00:52.017
You could say this triangle ADC.

00:00:53.680 --> 00:00:56.882
It has it stacked on
top of, its base is the

00:00:57.131 --> 00:01:01.085
hypotenuse of triangle
ACD, which I could, I'm

00:01:01.085 --> 00:01:03.866
going to outline it in
blue since I already

00:01:03.866 --> 00:01:07.584
labelled the measure of
this angle as being y.

00:01:11.572 --> 00:01:16.649
AC which is the base
of triangle ADC is the

00:01:17.242 --> 00:01:21.507
hypotenuse of triangle
ABC. They're stacked on

00:01:21.507 --> 00:01:24.740
top of each other like
that, just like that.

00:01:25.225 --> 00:01:27.275
The way I'm going to think
about it is first if you just

00:01:27.275 --> 00:01:31.130
look at this what is sine
of x plus y going to be?

00:01:31.756 --> 00:01:34.914
Well x plus y is this entire
angle right over here.

00:01:34.914 --> 00:01:40.209
If you look at this right
triangle, right triangle ADF,

00:01:40.209 --> 00:01:42.805
we know that the sine of an angle is

00:01:42.805 --> 00:01:45.116
the opposite side over the hypotenuse.

00:01:45.366 --> 00:01:47.349
Well the hypotenuse here
is one so the sine of this

00:01:47.349 --> 00:01:50.775
is the opposite over one,
or the sine of this angle,

00:01:50.775 --> 00:01:53.209
the sine of x plus y is equal to

00:01:53.209 --> 00:01:55.099
the length of this opposite side.

00:01:55.323 --> 00:01:57.538
So sine of x plus y is going to be

00:01:57.538 --> 00:02:00.662
equal to the length of segment DF.

00:02:03.795 --> 00:02:06.623
What I'm going to try to do
is, okay, length of segment DF

00:02:06.826 --> 00:02:08.685
is essentially what we're
looking for but we can

00:02:08.685 --> 00:02:12.813
decompose the length of
segment DF into two segments.

00:02:12.813 --> 00:02:15.763
We can decompose it into length of

00:02:15.763 --> 00:02:21.295
segment DE and the length of segment EF.

00:02:22.170 --> 00:02:24.450
We can say that DF, which
is the same thing as

00:02:24.450 --> 00:02:27.169
sine of x plus y, the length of segment DF

00:02:27.169 --> 00:02:30.171
is the same thing, is
equal to the length of

00:02:30.171 --> 00:02:36.565
segment DE plus the length of segment EF.

00:02:39.724 --> 00:02:44.626
EF is of course the same thing
as the length of segment CB.

00:02:44.626 --> 00:02:47.516
ECBF, this right over here is a

00:02:47.516 --> 00:02:50.018
rectangle so EF is the same thing as CB.

00:02:50.343 --> 00:02:57.251
So this thing is going to be
equal to DE right over here,

00:02:57.251 --> 00:03:03.958
length of segment DE plus
the length of segment CB.

00:03:07.358 --> 00:03:09.422
Once again the way I'm
going to address this,

00:03:09.422 --> 00:03:13.094
the sine of x plus y which
is the length of DF and DF

00:03:13.094 --> 00:03:17.203
can be decomposed as the
lengths of DE and CB.

00:03:18.016 --> 00:03:21.531
Now with that as a hint I
encourage you to figure out what

00:03:21.531 --> 00:03:27.140
the length of segment DE is
in terms of x's and y's and

00:03:27.140 --> 00:03:30.359
sines and cosines, and also
figure out what the length of

00:03:30.359 --> 00:03:34.735
segment CB is in terms of x's
and y's and sines and cosines.

00:03:34.735 --> 00:03:36.271
Try to figure out as much as you can about

00:03:36.271 --> 00:03:39.474
this and these two might fall out of that.

00:03:40.612 --> 00:03:43.909
I'm assuming you've given
a go at it so now that we

00:03:43.909 --> 00:03:46.862
know that sine of x plus y
can be expressed this way,

00:03:46.862 --> 00:03:48.988
let's see if we can
figure these things out.

00:03:48.988 --> 00:03:50.801
I'm going to try to
address it by figuring out

00:03:50.801 --> 00:03:54.097
as many lengths and angles here as I can.

00:03:54.786 --> 00:03:58.988
Let's go to this top red
triangle right over here.

00:03:59.613 --> 00:04:02.176
Its hypotenuse has length one so what's

00:04:02.176 --> 00:04:04.004
going to be the length of segment DC?

00:04:04.800 --> 00:04:08.396
That is the opposite
side of our angle x so

00:04:08.396 --> 00:04:12.400
we know sine of x is equal to DC over one,

00:04:12.676 --> 00:04:14.427
or DC over one is just DC.

00:04:14.894 --> 00:04:17.910
This length right over here is sine of x.

00:04:18.487 --> 00:04:21.477
Segment AC, same exact logic.

00:04:21.477 --> 00:04:24.801
Cosine of x is the length of AC over

00:04:24.801 --> 00:04:26.144
one which is just the length of AC.

00:04:26.144 --> 00:04:28.441
This length right over here,

00:04:28.675 --> 00:04:33.784
segment AC its length is cosine of x.

00:04:34.566 --> 00:04:37.300
That's kind of interesting. Now
let's see what we can figure

00:04:37.300 --> 00:04:41.817
out about this triangle,
triangle ACB right over here.

00:04:42.966 --> 00:04:45.043
How could we figure out CB?

00:04:45.732 --> 00:04:49.845
Well we know that sine of
y, let me write this here.

00:04:49.845 --> 00:04:52.216
Sine of y is equal to what?

00:04:52.575 --> 00:04:58.294
It's equal to the length of
segment CB over the hypotenuse.

00:04:58.294 --> 00:05:01.075
The hypotenuse here is the cosine of x,

00:05:01.075 --> 00:05:03.436
and I think you might see
where all of this is leading.

00:05:03.436 --> 00:05:05.169
At any point if you get excited pause the

00:05:05.169 --> 00:05:07.888
video and try to finish
the proof on your own.

00:05:08.497 --> 00:05:10.546
The length of segment CB if we just

00:05:10.546 --> 00:05:13.689
multiply both sides by cosine of x,

00:05:14.121 --> 00:05:17.778
the length of segment CB is equal

00:05:17.778 --> 00:05:23.857
to cosine of x times sine of y.

00:05:27.966 --> 00:05:32.387
Which is neat because we
just showed that this thing

00:05:32.387 --> 00:05:36.154
right over here is equal to
this thing right over here.

00:05:36.390 --> 00:05:40.903
To complete our proof we
just need to prove that this

00:05:40.903 --> 00:05:46.108
thing is equal to this
thing right over there.

00:05:46.108 --> 00:05:47.717
If that's equal to that
and that's equal to that

00:05:47.717 --> 00:05:49.451
well we already know that
the sum of these is equal

00:05:49.451 --> 00:05:53.169
to the length of DF which
is sine of x plus y.

00:05:53.812 --> 00:05:57.950
Let's see if we can figure out,
if we can express DE somehow.

00:05:59.122 --> 00:06:02.391
What angle would be useful?
If somehow we could figure

00:06:02.391 --> 00:06:06.643
out this angle up here or maybe
this angle, well let's see.

00:06:06.643 --> 00:06:12.873
If we could figure out
this angle then DE we could

00:06:12.873 --> 00:06:15.297
express in terms of this
angle and sine of x.

00:06:15.297 --> 00:06:16.638
Let's see if we can figure out that angle.

00:06:16.905 --> 00:06:21.497
We know this is angle y over here and

00:06:21.497 --> 00:06:24.825
we also know that this is a right angle.

00:06:25.278 --> 00:06:30.700
EC is parallel to AB so you
could view AC as a transversal.

00:06:31.538 --> 00:06:35.166
If this is angle y right over here

00:06:35.166 --> 00:06:38.493
then we know this is also angle y.

00:06:39.429 --> 00:06:43.571
These are once again, notice.
If AC is a transversal here

00:06:43.571 --> 00:06:47.010
and EC and AB are parallel then
if this is y then that is y.

00:06:48.557 --> 00:06:51.369
If that's y then this is 90 minus y.

00:06:54.608 --> 00:06:58.838
If this is 90 degrees and
this is 90 minus y then

00:06:58.838 --> 00:07:03.372
these two angles combined
add up to 180 minus y,

00:07:03.997 --> 00:07:05.761
and if all three of these add up to 180

00:07:05.761 --> 00:07:08.734
then this thing up here
must be equal to y.

00:07:08.734 --> 00:07:13.122
Validate that. y plus 90 minus
y plus 90 is 180 degrees,

00:07:13.447 --> 00:07:16.510
and that is useful for
us because now we can

00:07:16.510 --> 00:07:21.682
express segment DE in
terms of y and sine of x.

00:07:22.136 --> 00:07:24.557
What is DE to y? It's the adjacent

00:07:24.557 --> 00:07:26.450
angle, so we can think of cosine.

00:07:27.088 --> 00:07:30.963
We know that the cosine
of angle y, if we look at

00:07:30.963 --> 00:07:35.733
triangle DEC right over here,
we know that the cosine of y

00:07:35.733 --> 00:07:45.058
is equal to segment DE over
its hypotenuse, over sine of x.

00:07:46.119 --> 00:07:48.432
You should be getting excited
right about now because we've

00:07:48.823 --> 00:07:53.089
just shown, if we multiply
both sides by sine of x,

00:07:53.089 --> 00:07:56.072
we've just shown that DE is equal

00:07:56.072 --> 00:08:01.665
to sine of x times cosine of y.

00:08:03.529 --> 00:08:08.885
We've now shown that
this is equal to this.

00:08:09.276 --> 00:08:12.010
We already showed that CB
is equal to that, so the sum

00:08:12.010 --> 00:08:15.809
of DE and CB which is the
same thing as the sum of DE

00:08:15.809 --> 00:08:20.730
and EF is the sine of x plus
y which is that over there.

00:08:20.730 --> 00:08:22.699
We are done, we have proven the

00:08:22.699 --> 00:08:25.059
angle addition formula for sine.


Sine of non special angle

WEBVTT
Kind: captions
Language: en

00:00:00.567 --> 00:00:02.366
Voiceover:What I want to
attempt to do in this video

00:00:02.366 --> 00:00:07.236
is figure out what the
sine of seven pi over 12 is

00:00:07.236 --> 00:00:10.100
without using a calculator.

00:00:10.100 --> 00:00:12.966
And so let's just
visualize seven pi over 12

00:00:12.966 --> 00:00:14.601
in the unit circle.

00:00:15.034 --> 00:00:18.300
One side of the angle is going
along the positive x-axis

00:00:18.300 --> 00:00:18.966
and then let's see,

00:00:18.966 --> 00:00:20.837
if we go straight up, that's pi over two,

00:00:20.837 --> 00:00:22.767
which is the same thing as six pi over 12,

00:00:22.767 --> 00:00:25.234
so then we essentially just
have another pi over 12

00:00:25.234 --> 00:00:27.036
to get right over there.

00:00:27.036 --> 00:00:29.234
This is the angle that we're talking about

00:00:29.234 --> 00:00:33.034
that is seven pi over 12 radians,

00:00:33.034 --> 00:00:33.967
and the sine of it,

00:00:33.967 --> 00:00:36.367
by the unit circle definition of sine,

00:00:36.982 --> 00:00:39.356
it's the y-coordinate of where this ray

00:00:39.356 --> 00:00:40.966
intersects the unit circle.

00:00:40.966 --> 00:00:42.167
This is the unit circle,

00:00:42.167 --> 00:00:43.366
has radius one

00:00:43.366 --> 00:00:44.766
where it intersects the unit.

00:00:44.766 --> 00:00:46.633
The y-coordinate is the sine.

00:00:46.633 --> 00:00:48.033
Another way to think about it,

00:00:48.033 --> 00:00:52.834
it's the length of this
line right over here.

00:00:52.834 --> 00:00:55.567
I encourage you to pause
the video right now

00:00:55.567 --> 00:00:56.567
and try to think about it on your own.

00:00:56.567 --> 00:00:59.501
See if you can use your
powers of trigonometry

00:00:59.501 --> 00:01:02.902
to figure out what sine
of seven pi over 12 is

00:01:02.902 --> 00:01:05.333
or essentially the length
of this magenta line.

00:01:05.700 --> 00:01:07.848
I'm assuming you've given a go at it,

00:01:07.848 --> 00:01:09.633
and if you're like me,

00:01:09.633 --> 00:01:11.033
your first temptation might have been

00:01:11.033 --> 00:01:13.181
just to focus on this
triangle right over here

00:01:13.181 --> 00:01:15.033
that I drew for you.

00:01:15.033 --> 00:01:17.576
The triangle looks like this.

00:01:18.115 --> 00:01:19.834
It looks like this,

00:01:19.834 --> 00:01:23.100
where that's what you're
trying to figure out,

00:01:23.100 --> 00:01:24.034
this length right over here,

00:01:24.034 --> 00:01:26.510
sine of seven pi over 12.

00:01:26.730 --> 00:01:32.134
We know the length of
the hypotenuse is one.

00:01:32.500 --> 00:01:34.633
It's a radius of the unit circle.

00:01:34.633 --> 00:01:36.833
It's a right triangle right over there.

00:01:36.833 --> 00:01:39.500
We also know this angle right over here,

00:01:39.500 --> 00:01:42.233
which is this angle right over here,

00:01:42.233 --> 00:01:43.566
this gets us six pi over 12,

00:01:43.566 --> 00:01:44.633
and then we have another pi over 12,

00:01:44.633 --> 00:01:47.233
so we know that that is pi over 12,

00:01:47.233 --> 00:01:48.433
not pi over 16.

00:01:48.433 --> 00:01:51.266
We know that this angle right
over here is pi over 12.

00:01:51.567 --> 00:01:52.700
Given this information,

00:01:52.700 --> 00:01:54.499
we can figure out this,

00:01:54.499 --> 00:01:58.300
or we can at least relate
this side to this other side

00:01:58.300 --> 00:02:00.767
using a trig function
relative to this angle.

00:02:00.767 --> 00:02:02.800
This is the adjacent side.

00:02:03.833 --> 00:02:05.834
The cosine of pi over 12 is going to be

00:02:05.834 --> 00:02:07.766
this magenta side over one,

00:02:07.766 --> 00:02:09.700
or you could just say it's
equal to this magenta side.

00:02:09.700 --> 00:02:13.334
You could say that this
is cosine of pi over 12.

00:02:13.766 --> 00:02:16.100
We just figured out that
sine of seven pi over 12

00:02:16.100 --> 00:02:18.967
is the same thing as cosine of pi over 12,

00:02:18.967 --> 00:02:20.100
but that still doesn't help me.

00:02:20.100 --> 00:02:21.100
I don't know offhand what the

00:02:21.100 --> 00:02:22.767
cosine of pi over 12 radians is

00:02:22.767 --> 00:02:24.400
without using a calculator.

00:02:24.991 --> 00:02:27.300
Instead of thinking about it this way,

00:02:27.300 --> 00:02:29.570
let's see if we can compose this angle

00:02:30.000 --> 00:02:32.500
or if we can decompose it into some angles

00:02:32.500 --> 00:02:34.566
for which we do know the sine and cosine.

00:02:34.566 --> 00:02:35.633
What angles are those?

00:02:35.633 --> 00:02:37.953
Those are the angles in
special right triangles.

00:02:38.730 --> 00:02:39.566
For example,

00:02:39.566 --> 00:02:42.968
we are very familiar
with 30-60-90 triangles.

00:02:42.968 --> 00:02:46.670
30-60-90 triangles look
something like this.

00:02:47.115 --> 00:02:49.934
This is my best attempt
at hand drawing it.

00:02:51.992 --> 00:02:53.566
Instead of writing 30-degree side,

00:02:53.566 --> 00:02:54.700
since we're thinking in radians,

00:02:54.700 --> 00:02:57.071
I'll write that as pi over six radians.

00:02:57.366 --> 00:02:59.033
The 60-degree side,
I'm going to write that

00:02:59.033 --> 00:03:00.547
as pi over three radians,

00:03:00.547 --> 00:03:02.701
and of course, this is the right angle.

00:03:02.701 --> 00:03:05.501
If the hypotenuse here is one,

00:03:05.501 --> 00:03:07.773
then the side opposite the 30-degree side,

00:03:07.773 --> 00:03:09.233
or the pi over six radian side,

00:03:09.233 --> 00:03:10.501
is going to be half the hypotenuse,

00:03:10.501 --> 00:03:12.367
which, in this case is 1/2.

00:03:12.367 --> 00:03:15.233
Then the other side that's
opposite the 60-degree side

00:03:15.233 --> 00:03:17.501
or the pi over three radian side,

00:03:17.501 --> 00:03:19.779
is going to be square root of
three times the shorter side.

00:03:19.779 --> 00:03:22.701
It's going to be square
root of three over two.

00:03:22.701 --> 00:03:24.978
We've used these types
of triangles in the past

00:03:24.978 --> 00:03:28.100
to figure out the sine
or cosine of 30 or 60,

00:03:28.100 --> 00:03:29.100
or in this case,

00:03:29.100 --> 00:03:31.900
pi over six or pi over three.

00:03:31.900 --> 00:03:34.104
We know about pi over
six and pi over three.

00:03:34.104 --> 00:03:37.366
We also know about 45-45-90 triangles.

00:03:37.366 --> 00:03:42.036
We know that they're
isosceles right triangles.

00:03:42.036 --> 00:03:43.434
They look like this,

00:03:43.434 --> 00:03:45.300
my best attempt at drawing it.

00:03:45.300 --> 00:03:47.033
That one actually doesn't
look that isosceles,

00:03:47.033 --> 00:03:50.799
so let me make it a little bit more ...

00:03:51.592 --> 00:03:52.176
I don't know.

00:03:52.176 --> 00:03:57.533
That looks closer to being
an isosceles right triangle.

00:03:58.166 --> 00:04:00.966
We know if the length of
the hypotenuse is one,

00:04:00.966 --> 00:04:02.967
and this comes straight
out of Pythagorean theorem,

00:04:02.967 --> 00:04:04.966
then the length of each
of the other two sides

00:04:04.966 --> 00:04:07.433
are going to be square
root of two over two

00:04:07.433 --> 00:04:08.233
times the hypotenuse,

00:04:08.233 --> 00:04:10.433
which, in this case, is the
square root of two over two.

00:04:10.433 --> 00:04:12.366
Instead of describing
these as 45-degree angles,

00:04:12.366 --> 00:04:15.500
we know that's the same
thing as pi over four,

00:04:15.500 --> 00:04:18.266
pi over four radians.

00:04:18.900 --> 00:04:21.367
If you give me pi over six,
pi over three, pi over four,

00:04:21.367 --> 00:04:22.900
I can use these triangles

00:04:22.900 --> 00:04:24.567
either using the classic definition,

00:04:24.567 --> 00:04:25.766
SOHCAHTOA definitions,

00:04:25.766 --> 00:04:28.300
or I could stick them
on the unit circle here

00:04:28.300 --> 00:04:30.033
to use the unit circle
definition of trig functions

00:04:30.033 --> 00:04:32.115
to figure out what the
sine, cosine, or tangent

00:04:32.115 --> 00:04:34.039
of these angles are.

00:04:34.039 --> 00:04:37.366
Can I decompose seven pi
over 12 into some combination

00:04:37.366 --> 00:04:41.133
of pi over sixes, pi over
threes, or pi over fours?

00:04:41.366 --> 00:04:42.166
Think about that.

00:04:42.166 --> 00:04:43.968
Let me rewrite pi over six, pi over three,

00:04:43.968 --> 00:04:44.900
and pi over four

00:04:44.900 --> 00:04:46.766
with a denominator over 12.

00:04:46.766 --> 00:04:47.566
Let me write that.

00:04:47.566 --> 00:04:52.633
Pi over six is equal to two pi over 12,

00:04:52.633 --> 00:04:57.500
pi over three is equal to four pi over 12,

00:04:57.500 --> 00:05:04.266
and pi over four is equal
to three pi over 12.

00:05:04.434 --> 00:05:05.233
Let's see.

00:05:05.233 --> 00:05:07.033
Two plus four is not seven,

00:05:07.033 --> 00:05:08.367
two plus three is not seven,

00:05:08.367 --> 00:05:10.846
but four plus three is seven.

00:05:10.984 --> 00:05:15.166
So I could use this and this.

00:05:15.166 --> 00:05:17.436
Four pi over 12 plus three pi over 12

00:05:17.436 --> 00:05:20.166
is seven pi over 12.

00:05:20.166 --> 00:05:21.233
I could rewrite this.

00:05:21.233 --> 00:05:27.980
This is the same thing as
sine of three pi over 12

00:05:27.980 --> 00:05:32.700
plus 4 pi over 12,

00:05:32.700 --> 00:05:34.700
which, of course, is the same thing,

00:05:34.700 --> 00:05:37.779
sine of pi over 4,

00:05:37.779 --> 00:05:39.100
I'll do this in another color,

00:05:39.100 --> 00:05:42.933
sine of pi over 4 plus ...

00:05:44.633 --> 00:05:45.766
let me do this ...

00:05:45.766 --> 00:05:51.026
plus pi over three,

00:05:51.368 --> 00:05:55.368
Now we can use our angle
addition formula for sine

00:05:55.368 --> 00:05:57.971
in order to write this
as the sum of products

00:05:57.971 --> 00:06:00.900
of cosines and sines of these angles.

00:06:00.900 --> 00:06:03.174
Let's actually do that.

00:06:03.174 --> 00:06:06.300
This right over here is
going to be equal to,

00:06:06.300 --> 00:06:10.033
this is going to be equal to the sine,

00:06:10.033 --> 00:06:22.001
the sine of pi over four times
the cosine of pi over three

00:06:22.001 --> 00:06:24.237
plus the other way around,

00:06:24.237 --> 00:06:30.900
cosine of pi over four times
the sine of pi over three,

00:06:30.900 --> 00:06:33.233
sine of pi over three.

00:06:33.233 --> 00:06:34.833
Now we just have to
figure out these things,

00:06:34.833 --> 00:06:36.833
and I've already set up
the triangles to do it.

00:06:36.833 --> 00:06:39.357
What is sine of pi over four?

00:06:39.357 --> 00:06:40.500
Sine of pi over four,

00:06:40.500 --> 00:06:42.401
well, let's think about ...

00:06:42.633 --> 00:06:44.366
This is pi over four right over here.

00:06:44.366 --> 00:06:48.066
Sine is opposite over hypotenuse.

00:06:48.100 --> 00:06:51.067
That's just going to be
square root of two over two.

00:06:51.100 --> 00:06:53.379
This is square root of two over two,

00:06:53.379 --> 00:06:55.234
square root of two over two.

00:06:55.234 --> 00:06:57.900
What is cosine of pi over three?

00:06:57.900 --> 00:07:00.566
This is a pi over three
radian angle right over here.

00:07:00.566 --> 00:07:04.556
Cosine is adjacent over a hypotenuse.

00:07:05.632 --> 00:07:07.445
It's adjacent over a hypotenuse,

00:07:07.445 --> 00:07:09.852
so this is going to be 1/2.

00:07:09.972 --> 00:07:12.234
What is cosine of pi over four?

00:07:12.234 --> 00:07:13.300
Go back to pi over four.

00:07:13.300 --> 00:07:15.302
It's adjacent over a hypotenuse.

00:07:15.302 --> 00:07:16.967
It's square root of two over two.

00:07:16.967 --> 00:07:20.903
It is also square root of two over two,

00:07:20.903 --> 00:07:23.301
square root of two over two.

00:07:23.301 --> 00:07:25.833
What's sine of pi over three?

00:07:25.833 --> 00:07:28.966
Sine is opposite over a hypotenuse,

00:07:28.966 --> 00:07:31.100
so square root of three over two over one.

00:07:31.100 --> 00:07:33.500
Square root of three
over two divided by one,

00:07:33.500 --> 00:07:35.234
which is square root of three over two.

00:07:35.234 --> 00:07:38.701
Now we just have to simplify
all of this business.

00:07:38.701 --> 00:07:43.501
This is going to be
equal to the sum of this,

00:07:43.870 --> 00:07:45.700
or the product, I should say,

00:07:45.700 --> 00:07:49.966
is just square root of two over four,

00:07:49.966 --> 00:07:54.033
and then plus the product of these.

00:07:54.340 --> 00:07:55.033
Let's see.

00:07:55.033 --> 00:08:00.433
We could write that as
square root of six over four,

00:08:00.433 --> 00:08:03.767
square root of six over four,

00:08:03.767 --> 00:08:06.566
or we could just rewrite
this whole thing as,

00:08:06.566 --> 00:08:12.302
and we deserve a little bit
of a drum roll at this point,

00:08:12.302 --> 00:08:13.833
this is equivalent to,

00:08:13.833 --> 00:08:15.767
let me just scroll over
to the right a little bit.

00:08:15.767 --> 00:08:18.501
This is equivalent to square root of two

00:08:18.501 --> 00:08:21.980
plus square root of six,

00:08:21.980 --> 00:08:25.977
all of that over four.

00:08:25.977 --> 00:08:29.704
That's what sine of seven pi over 12 is,

00:08:29.704 --> 00:08:31.248
or cosine of pi over 12,

00:08:31.248 --> 00:08:33.333
what that is equal to.


Another example using angle addition formula with cosine

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:03.823
Voiceover:We're told theta
is between pi and 2pi,

00:00:03.823 --> 00:00:04.950
and cosine of theta is equal

00:00:04.950 --> 00:00:06.833
to negative square root of 3 over 2,

00:00:06.833 --> 00:00:09.767
and phi is an acute angle,

00:00:09.767 --> 00:00:12.700
and we can assume it's
a positive acute angle.

00:00:12.700 --> 00:00:14.967
So we could say an acute positive angle

00:00:14.967 --> 00:00:19.835
or as a positive acute angle.

00:00:19.835 --> 00:00:24.169
And cosine of phi is equal to 7/25.

00:00:24.169 --> 00:00:27.900
Find cosine of phi plus theta exactly.

00:00:27.900 --> 00:00:30.434
So essentially, can we figure
it out without a calculator?

00:00:30.434 --> 00:00:32.100
I encourage you to pause this video

00:00:32.100 --> 00:00:34.366
and think about it on your own.

00:00:34.366 --> 00:00:35.900
Let's see if we can work through it.

00:00:35.900 --> 00:00:38.367
When we see, "Find cosine
of phi plus theta,"

00:00:38.367 --> 00:00:42.233
we're finding the cosine of
the addition of 2 angles,

00:00:42.233 --> 00:00:43.234
so to me at least,

00:00:43.234 --> 00:00:46.233
that screams out that maybe
the angle addition formula

00:00:46.233 --> 00:00:48.900
can help us evaluate this,

00:00:48.900 --> 00:00:50.834
especially because we know
what cosine of theta is,

00:00:50.834 --> 00:00:52.033
cosine of phi is,

00:00:52.033 --> 00:00:52.900
and then maybe we can also use those

00:00:52.900 --> 00:00:55.433
to figure out what sine of
theta and sine of phi are.

00:00:55.433 --> 00:00:58.034
So let's just write out
the angle addition formula.

00:00:58.034 --> 00:01:08.167
It tells us that cosine of phi plus theta

00:01:08.167 --> 00:01:11.367
is equal to cosine of
both of those angles,

00:01:11.367 --> 00:01:14.033
the product of the cosines
of both of those angles.

00:01:14.033 --> 00:01:19.961
So cosine phi times cosine theta

00:01:19.961 --> 00:01:21.196
minus,

00:01:21.196 --> 00:01:23.101
so if this was a positive,
this is going to be a negative,

00:01:23.101 --> 00:01:25.033
if this was a negative,
this would be a positive,

00:01:25.033 --> 00:01:28.566
minus the product of the
sines of both of these angles,

00:01:28.566 --> 00:01:33.600
so sine of phi times sine of theta.

00:01:36.701 --> 00:01:38.833
And we already know some
of this information.

00:01:38.833 --> 00:01:40.900
We know what cosine of phi is.

00:01:40.900 --> 00:01:43.041
Cosine of phi is 7/25.

00:01:43.041 --> 00:01:45.433
So that is 7/25.

00:01:45.433 --> 00:01:47.034
We know what cosine of theta is.

00:01:47.034 --> 00:01:49.636
Cosine of theta is negative
square root of 3 over 2.

00:01:49.636 --> 00:01:52.434
Negative square root of 3 over 2,

00:01:52.434 --> 00:01:54.366
so we're going to take a
product here for this term.

00:01:54.366 --> 00:01:55.900
Now we need to figure out what sine of phi

00:01:55.900 --> 00:01:57.766
and sine of theta are.

00:01:57.766 --> 00:02:00.500
Lucky for us, we have
the Pythagorean identity.

00:02:00.500 --> 00:02:04.500
The Pythagorean identity tells
us that sine squared theta

00:02:04.500 --> 00:02:07.900
plus cosine squared theta is equal to 1.

00:02:07.900 --> 00:02:10.966
Or we could say that sine squared theta

00:02:10.966 --> 00:02:14.166
is equal to 1 minus cosine squared theta,

00:02:14.166 --> 00:02:19.967
or that sine of theta is
equal to the plus or minus

00:02:19.967 --> 00:02:25.033
square root of 1 minus
cosine squared theta.

00:02:25.033 --> 00:02:27.100
For example, we could use
this now to figure out

00:02:27.100 --> 00:02:29.034
what sine of theta is.

00:02:29.034 --> 00:02:33.566
We could say sine of
theta is going to be equal

00:02:33.566 --> 00:02:35.833
to the plus or minus square root

00:02:35.833 --> 00:02:39.566
of 1 minus cosine squared theta.

00:02:39.566 --> 00:02:42.566
Cosine squared theta is negative
square root of 3 over 2.

00:02:42.566 --> 00:02:44.701
If you square it, that's
going to be positive,

00:02:44.701 --> 00:02:46.905
and if you square 3,

00:02:46.905 --> 00:02:48.633
if you square the square root of 3,

00:02:48.633 --> 00:02:50.366
you're going to get 3,

00:02:50.366 --> 00:02:52.934
and if you square 2,
you're going to get 4.

00:02:54.525 --> 00:02:56.646
The plus or minus square
root of 1 minus 3/4,

00:02:56.646 --> 00:03:01.432
which is equal to the plus
or minus square root of 1/4,

00:03:01.432 --> 00:03:05.600
which is equal to plus or minus 1/2.

00:03:06.100 --> 00:03:07.433
Now which one is it going to be?

00:03:07.433 --> 00:03:12.433
Is sine of theta going to be
positive 1/2 or negative 1/2?

00:03:12.433 --> 00:03:13.233
To think about that,

00:03:13.233 --> 00:03:17.600
we could draw ourselves a
little unit circle here.

00:03:17.909 --> 00:03:19.900
That's my Y-axis.

00:03:19.900 --> 00:03:24.167
That is my X-axis.

00:03:24.167 --> 00:03:26.684
Let me draw a little unit circle here,

00:03:26.684 --> 00:03:29.500
as neatly as I can.

00:03:29.500 --> 00:03:31.500
A little unit circle right over there.

00:03:31.500 --> 00:03:33.238
Now what do they tell us about theta?

00:03:33.238 --> 00:03:37.900
They tell us that theta
is between pi and 2 pi,

00:03:37.900 --> 00:03:40.700
so it's between pi and 2 pi.

00:03:40.700 --> 00:03:44.233
So our angle, our terminal,

00:03:44.233 --> 00:03:46.166
I guess the terminal ray of the angle

00:03:46.166 --> 00:03:47.367
is going to sit,

00:03:47.367 --> 00:03:50.233
is going to be in the
third or fourth quadrants.

00:03:50.233 --> 00:03:52.033
We're saying sine of theta is equal

00:03:52.033 --> 00:03:54.234
either positive 1/2 or negative 1/2,

00:03:54.234 --> 00:03:55.966
so it's either positive 1/2,

00:03:55.966 --> 00:03:59.367
which could mean it's one of
these angles right over here,

00:03:59.367 --> 00:04:00.633
or it's negative 1/2,

00:04:00.633 --> 00:04:03.900
which means it's one of
these angles right over here.

00:04:03.900 --> 00:04:06.234
This tells us that we're in
the third or fourth quadrant,

00:04:06.234 --> 00:04:07.033
so sine of theta,

00:04:07.033 --> 00:04:10.033
we don't know if theta is this angle

00:04:10.033 --> 00:04:12.835
or if theta is this angle right over here,

00:04:12.835 --> 00:04:15.300
but we know if it's in the
third or fourth quadrant,

00:04:15.300 --> 00:04:18.234
the sine of it is going
to be non-positive.

00:04:18.664 --> 00:04:20.833
We know that for this theta,

00:04:20.833 --> 00:04:25.900
sine of theta is going to be negative 1/2,

00:04:25.900 --> 00:04:27.500
negative 1/2.

00:04:27.500 --> 00:04:31.666
So this right over here is negative 1/2.

00:04:31.701 --> 00:04:34.166
Now let's think about sine of phi.

00:04:34.166 --> 00:04:37.836
Sine of phi is going to be equal

00:04:37.836 --> 00:04:40.042
to plus or minus square root

00:04:40.042 --> 00:04:44.034
of 1 minus cosine of phi squared.

00:04:44.034 --> 00:04:45.700
Cosine of phi is 7/25,

00:04:45.700 --> 00:04:51.802
so that's 49 over 625.

00:04:51.925 --> 00:04:52.301
Let's see.

00:04:52.301 --> 00:04:53.901
What is that going to be?

00:04:53.901 --> 00:04:54.833
Let me do it over here.

00:04:54.833 --> 00:05:01.635
625 over 625 minus 49 over 625,

00:05:01.635 --> 00:05:04.007
I just rewrote 1 as 625 over 625.

00:05:04.007 --> 00:05:07.915
That's going to be 625
minus 50 would be 575.

00:05:07.915 --> 00:05:08.566
That's going to be one more.

00:05:08.566 --> 00:05:13.367
That's going to be 576 over 625,

00:05:13.367 --> 00:05:15.900
so it's equal to the
plus or minus square root

00:05:15.900 --> 00:05:20.166
of 576 over 625.

00:05:20.566 --> 00:05:21.100
And let's see.

00:05:21.100 --> 00:05:22.766
I know what the square root of 625 is.

00:05:22.766 --> 00:05:23.900
It's 25.

00:05:23.900 --> 00:05:26.368
576 , is it 24?

00:05:26.368 --> 00:05:30.634
24 times 24, yup, it is 576.

00:05:30.634 --> 00:05:34.866
So this is equal to the
plus or minus 24/25.

00:05:34.900 --> 00:05:37.633
So sine of phi is 24/25.

00:05:37.633 --> 00:05:42.101
Remember, the sine of an
angle is the Y-coordinate

00:05:42.101 --> 00:05:44.701
of where the terminal ray
intersects the unit circle.

00:05:45.008 --> 00:05:49.034
So we're either looking
at one of these angles.

00:05:49.034 --> 00:05:50.849
We're either looking
at one of those angles,

00:05:50.849 --> 00:05:53.166
if the sine is a positive,

00:05:53.426 --> 00:05:56.233
so we're either looking at
this angle or that angle,

00:05:56.233 --> 00:06:00.100
or we're looking at a
terminal ray down here again.

00:06:00.100 --> 00:06:04.166
Now they tell us that phi
is a positive acute angle.

00:06:04.166 --> 00:06:07.633
So we know that we're dealing
actually with this scenario,

00:06:07.633 --> 00:06:09.433
this scenario right over here.

00:06:09.433 --> 00:06:13.038
Sine of phi is going to
be the positive 24/25,

00:06:13.038 --> 00:06:15.833
so that's 24/25.

00:06:15.833 --> 00:06:18.300
Now we just have to multiply the numbers

00:06:18.300 --> 00:06:19.766
and then do the subtraction.

00:06:19.766 --> 00:06:23.166
This is going to be equal to 725 ...

00:06:23.166 --> 00:06:24.366
let me just write it down.

00:06:24.366 --> 00:06:33.101
This is going to be equal to
negative 7 square roots of 3

00:06:33.101 --> 00:06:36.766
over 25 times 2 is 50,

00:06:36.766 --> 00:06:38.233
over 50,

00:06:38.233 --> 00:06:39.634
minus,

00:06:39.634 --> 00:06:40.930
but then we're going to
have a negative out here,

00:06:40.930 --> 00:06:42.371
so we could say plus.

00:06:42.494 --> 00:06:44.234
Negative times negative is positive.

00:06:44.234 --> 00:06:48.633
And then 24 over 25
times 1/2 is 12 over 25,

00:06:48.633 --> 00:06:52.433
so plus 12 over 25.

00:06:52.433 --> 00:06:53.833
But actually, let me just write it over 50

00:06:53.833 --> 00:06:55.300
since we have a 50 right over here.

00:06:55.300 --> 00:07:01.033
This is going to be plus 24 over 50.

00:07:01.033 --> 00:07:03.700
And so this is going to be equal to

00:07:03.700 --> 00:07:10.966
24 minus 7 times the
square root of 3 over 50.

00:07:10.966 --> 00:07:12.866
And we are done.


Applying angle addition formula for sin

WEBVTT
Kind: captions
Language: en

00:00:00.591 --> 00:00:01.858
Voiceover:What I want to do in this video

00:00:01.858 --> 00:00:04.364
is use all of our powers,
all of our knowledge

00:00:04.364 --> 00:00:06.298
of trig functions and trig identities.

00:00:06.298 --> 00:00:07.865
In order to figure out,

00:00:07.865 --> 00:00:10.294
given all the information
that we have here,

00:00:10.294 --> 00:00:13.706
in order to figure out the
length of this yellow line,

00:00:13.706 --> 00:00:16.207
this point, this line or the segment

00:00:16.207 --> 00:00:18.807
that goes from here to here.

00:00:18.807 --> 00:00:20.249
I encourage you to pause the video

00:00:20.249 --> 00:00:23.011
and think about it
before I work through it.

00:00:23.795 --> 00:00:25.780
I'm assuming you've had a go at it

00:00:25.780 --> 00:00:28.156
and in doing that you might have realized,

00:00:28.156 --> 00:00:31.172
okay this line that's one of the sides

00:00:31.172 --> 00:00:34.669
of this right triangle that
I have right over here.

00:00:34.669 --> 00:00:38.745
We're given this alpha and beta

00:00:38.745 --> 00:00:42.871
but if we consider the combined angle

00:00:42.871 --> 00:00:44.779
alpha plus beta,

00:00:44.779 --> 00:00:46.632
then this side right over here,

00:00:46.632 --> 00:00:48.511
we can just take out our
traditional trig functions.

00:00:48.511 --> 00:00:53.408
Our soh cah toa definition
of the basic trig functions

00:00:53.408 --> 00:00:56.593
and we know that sine is
opposite over hypotenuse.

00:00:56.593 --> 00:00:59.482
If we're considering alpha plus beta,

00:00:59.482 --> 00:01:01.011
this angle right over here,

00:01:01.011 --> 00:01:02.336
opposite over hypotenuse

00:01:02.336 --> 00:01:05.422
that's going to be this
length over the hypotenuse

00:01:05.422 --> 00:01:06.427
which is one.

00:01:06.427 --> 00:01:08.265
Sine of alpha plus beta

00:01:08.265 --> 00:01:10.848
is going to be this
length right over here.

00:01:10.848 --> 00:01:12.979
That seems interesting,
so let me write that down.

00:01:12.979 --> 00:01:21.249
Sine of alpha plus beta

00:01:23.525 --> 00:01:25.972
is essentially what we're looking for.

00:01:25.972 --> 00:01:27.636
Sine of alpha plus beta

00:01:27.636 --> 00:01:30.174
is this length right over here.

00:01:30.174 --> 00:01:31.676
Sine of alpha plus beta

00:01:31.676 --> 00:01:33.431
it's equal to the opposite side,

00:01:33.431 --> 00:01:35.107
that over the hypotenuse.

00:01:35.107 --> 00:01:37.129
Well the hypotenuse is just
going to be equal to one

00:01:37.129 --> 00:01:38.722
so it's equal to this side.

00:01:38.722 --> 00:01:41.737
Another way of raising
the exact same problem

00:01:41.737 --> 00:01:43.202
that we first tried to tackle is

00:01:43.202 --> 00:01:47.152
how do we figure out the
sine of alpha plus beta?

00:01:48.890 --> 00:01:51.185
If you're familiar with
your trig identities

00:01:51.185 --> 00:01:53.112
something might be jumping out of you.

00:01:53.112 --> 00:01:56.558
Hey, we know a different
way of expressing sine

00:01:56.558 --> 00:01:58.267
of alpha plus beta.

00:01:58.267 --> 00:02:01.064
We know that this thing is the same thing

00:02:01.064 --> 00:02:10.903
as the sine of alpha
times the cosine of beta

00:02:10.903 --> 00:02:14.068
plus the other way around,

00:02:14.068 --> 00:02:24.089
the cosine of alpha
times the sine of beta.

00:02:24.089 --> 00:02:26.222
Let me a draw a line here
so we don't get confused.

00:02:26.222 --> 00:02:28.381
If we're trying to figure this out,

00:02:28.381 --> 00:02:30.776
and we know that this can
be re-expressed this way

00:02:30.776 --> 00:02:33.050
it all boils down to can we figure out

00:02:33.050 --> 00:02:34.911
what sine of alpha is, cosine of beta,

00:02:34.911 --> 00:02:37.825
cosine of alpha, and sine of beta.

00:02:37.825 --> 00:02:39.391
Now when you look at this,

00:02:39.391 --> 00:02:42.064
you see that you actually
can figure those things out.

00:02:42.064 --> 00:02:43.010
Let's do that.

00:02:43.010 --> 00:02:45.374
Sine of alpha, I'll write it over here.

00:02:45.374 --> 00:02:48.811
Sine of alpha is equal to ...

00:02:48.811 --> 00:02:49.748
This is alpha,

00:02:49.748 --> 00:02:52.483
sine is opposite over hypotenuse.

00:02:52.483 --> 00:02:54.716
It's 0.5 over one.

00:02:54.716 --> 00:02:59.809
This is equal to 0.5, so that is 0.5.

00:02:59.809 --> 00:03:04.917
Cosine of beta

00:03:04.917 --> 00:03:06.254
this is beta.

00:03:06.254 --> 00:03:09.945
Cosine is adjacent over hypotenuse.

00:03:09.945 --> 00:03:12.824
This is beta, the adjacent side is 0.6

00:03:12.824 --> 00:03:14.488
over the hypotenuse of 1.

00:03:14.488 --> 00:03:18.565
It's 0.6.

00:03:20.534 --> 00:03:25.113
Cosine of alpha.

00:03:25.113 --> 00:03:26.721
Adjacent over hypotenuse,

00:03:26.721 --> 00:03:29.293
it's square root of
three over two, over one.

00:03:29.293 --> 00:03:31.869
That's just square root of three over two.

00:03:32.853 --> 00:03:37.546
This is just square
root of three, over two.

00:03:37.546 --> 00:03:43.034
Then finally sine of beta.

00:03:43.034 --> 00:03:46.799
Opposite over hypotenuse is 0.8.

00:03:46.799 --> 00:03:51.532
This is 0.8.

00:03:51.532 --> 00:03:53.385
Actually let me write that as,

00:03:53.385 --> 00:03:55.555
I'm going to write that as four-fifths

00:03:55.555 --> 00:03:59.405
just so that that's the same thing as 0.8.

00:03:59.405 --> 00:04:00.278
Just because I think

00:04:00.278 --> 00:04:01.708
it's going to make it a
little bit easier for me

00:04:01.708 --> 00:04:03.212
to simplify right over here.

00:04:03.212 --> 00:04:04.939
What is all of these equal to?

00:04:04.939 --> 00:04:06.868
Well this is going to be equal to

00:04:06.868 --> 00:04:10.068
0.5 times 0.6,

00:04:10.068 --> 00:04:13.084
this part right over here is .3,

00:04:15.868 --> 00:04:20.718
and square root of three
over two, times four-fifths

00:04:20.718 --> 00:04:22.770
well let's just multiply them.

00:04:22.770 --> 00:04:25.191
Four divided by two is two.

00:04:25.191 --> 00:04:27.863
It's two square roots of three over five.

00:04:27.863 --> 00:04:30.524
This is equal to or so

00:04:30.524 --> 00:04:35.042
plus two square roots of three over five.

00:04:35.042 --> 00:04:36.940
This is essentially our answer,

00:04:36.940 --> 00:04:39.021
I feel a little uncomfortable having it

00:04:39.021 --> 00:04:40.604
in these two different formats

00:04:40.604 --> 00:04:42.398
where I have a fraction here
and I have a decimal here

00:04:42.398 --> 00:04:43.560
so let me just write the whole thing

00:04:43.560 --> 00:04:45.242
as a rational expression.

00:04:45.242 --> 00:04:51.162
0.3 is obviously the same
thing as three-tenths.

00:04:51.162 --> 00:04:53.998
That's the same things as three over ten

00:04:53.998 --> 00:04:57.749
plus, now this if I want to write over ten

00:04:57.749 --> 00:05:01.693
this the same things as four
square roots of three over ten

00:05:01.693 --> 00:05:03.842
and of course, if we add these two

00:05:03.842 --> 00:05:11.893
we are going to get three plus
four square roots of three,

00:05:11.893 --> 00:05:16.466
all of that over ten

00:05:16.466 --> 00:05:18.626
and we are done.


Angle addition formula with cosine

WEBVTT
Kind: captions
Language: en

00:00:00.488 --> 00:00:03.305
Voiceover:Cosine of two theta is equal to C

00:00:03.305 --> 00:00:06.280
and theta is between zero and pi.

00:00:06.997 --> 00:00:11.492
Write a formula for sine
of theta in terms of C

00:00:11.492 --> 00:00:13.985
and so I encourage you to pause the video

00:00:13.985 --> 00:00:15.771
and try to figure this out on your own

00:00:15.771 --> 00:00:17.278
before I work through it.

00:00:18.031 --> 00:00:19.485
So assuming you had a go,

00:00:19.485 --> 00:00:21.484
so let's see if we can
work through it together.

00:00:21.698 --> 00:00:23.171
I'm going to get my scratchpad out,

00:00:23.171 --> 00:00:24.984
I have copied and pasted
the exact question

00:00:24.984 --> 00:00:27.402
right over here, and so
let's think about it.

00:00:27.685 --> 00:00:30.171
They're telling us that
cosine of two theta

00:00:30.171 --> 00:00:32.642
is equal to C, so let
me write it this way.

00:00:33.174 --> 00:00:38.198
C is equal to cosine of two theta.

00:00:38.630 --> 00:00:43.432
Now, using my knowledge of
angle addition formulas,

00:00:43.432 --> 00:00:47.820
we know for example that cosine of

00:00:47.820 --> 00:01:00.804
alpha plus beta is equal to cosine alpha

00:01:00.804 --> 00:01:10.423
cosine beta minus sine alpha sine beta.

00:01:18.179 --> 00:01:20.409
Now why would this be useful here?

00:01:20.760 --> 00:01:23.718
Well this is the sum of
just theta plus theta,

00:01:23.718 --> 00:01:26.361
and so I can rewrite this in terms of

00:01:26.361 --> 00:01:28.281
cosines and sines and maybe I can rewrite

00:01:28.281 --> 00:01:30.332
the cosines in terms of sines

00:01:30.332 --> 00:01:32.128
and then solve for the sines.

00:01:32.328 --> 00:01:33.216
So let's try to do that.

00:01:33.415 --> 00:01:36.165
So I can rewrite cosine of
two theta that's the same

00:01:36.165 --> 00:01:45.985
as cosine of theta plus
theta, is of course the same

00:01:45.985 --> 00:01:48.336
thing as two theta, and I can use the

00:01:48.336 --> 00:01:50.274
angle addition formula for cosine.

00:01:50.690 --> 00:01:58.191
This is going to be
equal to cosine of theta,

00:01:58.191 --> 00:02:02.652
times cosine of theta,
times cosine of theta,

00:02:02.652 --> 00:02:09.628
minus sine of theta, times sine of theta,

00:02:09.628 --> 00:02:12.999
time sine of theta, which is of course

00:02:12.999 --> 00:02:18.090
equal to cosine squared
theta, that's equal to

00:02:18.090 --> 00:02:22.533
cosine squared theta,
minus and what we have

00:02:22.533 --> 00:02:24.932
right over here is sine squared theta.

00:02:27.815 --> 00:02:30.979
So let's see, we've been able to rewrite C

00:02:30.979 --> 00:02:33.014
in terms of cosine squared theta and

00:02:33.014 --> 00:02:34.732
sine squared theta, but ideally we just

00:02:34.732 --> 00:02:36.386
want to write it in terms of sine theta

00:02:36.386 --> 00:02:38.639
so we can solve for sine theta.

00:02:38.965 --> 00:02:43.085
So if we can re-express
cosine theta in terms of sine.

00:02:43.385 --> 00:02:45.501
Well we already know from
the pythagorean identity

00:02:45.501 --> 00:02:48.645
that cosine squared theta plus sine

00:02:48.645 --> 00:02:52.401
squared theta is equal
to one, or we could say

00:02:52.401 --> 00:02:55.576
that cosine squared theta is equal to,

00:02:55.576 --> 00:02:57.167
I'm just going to subtract
sine squared from both

00:02:57.167 --> 00:03:00.802
sides, is equal to one
minus sine squared theta.

00:03:00.999 --> 00:03:06.223
So let me rewrite this
as one minus sine squared

00:03:06.223 --> 00:03:09.241
theta, and then of course we have minus

00:03:09.241 --> 00:03:13.216
this yellow sine squared theta,

00:03:13.216 --> 00:03:19.906
and all of this is equal
to C or we could get that C

00:03:19.906 --> 00:03:27.404
is equal to one minus
two sine squared theta.

00:03:30.041 --> 00:03:31.682
And what's useful about this is we just

00:03:31.682 --> 00:03:34.677
have to solve for sine of theta.

00:03:35.110 --> 00:03:38.240
So let's see, I could
multiply both sides by a

00:03:38.240 --> 00:03:41.299
negative just so I can
switch the order over here.

00:03:41.455 --> 00:03:44.443
So I could write this as negative C

00:03:44.443 --> 00:03:49.361
is equal to two sine squared theta

00:03:49.361 --> 00:03:52.258
minus one and just multiply both sides

00:03:52.258 --> 00:03:55.370
by a negative and then
let's see I could add

00:03:55.370 --> 00:03:58.990
one to both sides, if I
add one to both sides,

00:03:58.990 --> 00:04:02.371
and I'll go over here, if
I add one to both sides,

00:04:02.371 --> 00:04:15.479
I get one minus C is equal
to two sine squared theta,

00:04:15.479 --> 00:04:20.427
I could divide both sides by two,

00:04:20.427 --> 00:04:22.139
and then so I get sine squared theta

00:04:22.139 --> 00:04:23.999
is equal to one minus C over two,

00:04:23.999 --> 00:04:27.081
or I could write that sine of theta

00:04:27.081 --> 00:04:30.034
is equal to the plus or minus square root

00:04:30.034 --> 00:04:32.758
of one minus C over two.

00:04:36.544 --> 00:04:38.392
So that leads to a question,

00:04:38.392 --> 00:04:39.356
Is it both?

00:04:39.356 --> 00:04:40.573
Is it the plus and minus square root?

00:04:40.573 --> 00:04:42.373
Or is it just one of those?

00:04:42.408 --> 00:04:43.711
And I encourage you to
pause the video again

00:04:43.711 --> 00:04:45.190
in case you haven't
already figured it out,

00:04:45.190 --> 00:04:46.550
and look at the information here,

00:04:46.550 --> 00:04:48.750
and think about whether
they give us the information

00:04:48.750 --> 00:04:50.173
of whether we should be looking

00:04:50.173 --> 00:04:51.957
at the positive or negative sine.

00:04:52.520 --> 00:04:57.248
Well they tell us that theta
is between zero and pi.

00:04:57.604 --> 00:04:59.871
So if I were to draw a unit circle here,

00:04:59.871 --> 00:05:01.701
between zero and pi radians.

00:05:02.372 --> 00:05:05.871
So, this angle right over
here is zero radians,

00:05:05.871 --> 00:05:08.092
and pi is going all the way over here.

00:05:10.411 --> 00:05:14.733
So this angle places its terminal ray

00:05:14.733 --> 00:05:16.686
either in the first or second quadrants.

00:05:17.257 --> 00:05:19.107
So, it could be an angle like this,

00:05:19.107 --> 00:05:20.792
it could be an angle like this,

00:05:20.792 --> 00:05:22.813
it cannot be an angle like this,

00:05:22.813 --> 00:05:27.363
and we know that the sine of an angle

00:05:27.363 --> 00:05:29.260
is the Y coordinate, and so we know that

00:05:29.260 --> 00:05:31.385
for the first of second
quadrant the Y coordinate

00:05:31.385 --> 00:05:33.431
is going to be non-negative.

00:05:33.731 --> 00:05:37.911
So we would want to take
the positive square root,

00:05:37.911 --> 00:05:42.635
so we would get sine of theta,

00:05:42.635 --> 00:05:45.448
is equal to the principal root,

00:05:45.448 --> 00:05:46.286
or we could even think of it

00:05:46.286 --> 00:05:47.482
as the positive square root

00:05:47.482 --> 00:05:50.547
of one minus C over two.

00:05:50.799 --> 00:05:52.278
So let's go back to our...

00:05:52.493 --> 00:05:54.532
Make sure we can check our answers.

00:05:54.865 --> 00:05:58.180
So sine of theta is
equal to the square root

00:05:58.180 --> 00:06:09.185
of one minus capital C,
all of that over two,

00:06:09.185 --> 00:06:10.923
and we got it right.


Inverse tan domain and range

WEBVTT
Kind: captions
Language: en

00:00:00.309 --> 00:00:02.206
Voiceover:We're told given g of x

00:00:02.206 --> 00:00:06.797
is equal to ten of x minus
three pi over two plus six,

00:00:06.797 --> 00:00:09.288
find the g inverse of x.

00:00:09.288 --> 00:00:10.952
They want us to type that in here

00:00:10.952 --> 00:00:12.699
and then they also want us to figure out

00:00:12.699 --> 00:00:15.264
what is the domain of g inverse,

00:00:15.264 --> 00:00:18.143
the domain of g inverse of x.

00:00:18.143 --> 00:00:20.715
I've got my little scratch pad here

00:00:20.715 --> 00:00:22.327
to try to work that through.

00:00:22.327 --> 00:00:25.387
Let's figure out what g inverse of x is.

00:00:25.387 --> 00:00:28.572
This is g of x, so g inverse of x.

00:00:28.572 --> 00:00:29.945
Essentially, let me just read

00:00:29.945 --> 00:00:31.862
this is g of x right over here,

00:00:31.862 --> 00:00:34.239
g of x is equal to tangent of x

00:00:34.239 --> 00:00:36.478
minus three pi over two plus six.

00:00:36.478 --> 00:00:39.997
G inverse of x, I
essentially can swap the …

00:00:39.997 --> 00:00:42.075
I can replace the x
with the g inverse of x

00:00:42.075 --> 00:00:43.809
and replace to g of x with an x

00:00:43.809 --> 00:00:46.805
and then solve for g inverse of x.

00:00:46.805 --> 00:00:58.908
I could write that x is equal
to tangent of g inverse of x

00:00:58.908 --> 00:01:05.863
minus three pi over two plus six.

00:01:05.863 --> 00:01:08.149
Let's just solve for g inverse of x.

00:01:08.149 --> 00:01:09.624
I actually encourage
you to pause this video

00:01:09.624 --> 00:01:12.584
and try to work through this
out or work it out on your own.

00:01:12.584 --> 00:01:15.015
Let's subtract six from both sides

00:01:15.015 --> 00:01:17.488
to at least get rid of this six here

00:01:17.488 --> 00:01:21.151
so I'm left with x minus
six is equal to the tangent

00:01:21.151 --> 00:01:28.702
of g inverse of x minus three pi over two.

00:01:28.702 --> 00:01:31.157
Now let's take the inverse tangent

00:01:31.157 --> 00:01:33.740
of both sides of this equation

00:01:33.740 --> 00:01:35.872
so the inverse tangent
on the left hand side

00:01:35.872 --> 00:01:40.487
is the inverse tangent of x minus six

00:01:40.487 --> 00:01:45.436
and on the right hand side the
inverse tangent of tangent.

00:01:45.436 --> 00:01:48.784
If we restrict the
domain in the proper way

00:01:48.784 --> 00:01:50.106
and we'll talk about that in a little bit

00:01:50.106 --> 00:01:51.943
is just going to be what the input

00:01:51.943 --> 00:01:53.771
into the tangent function is.

00:01:53.771 --> 00:01:56.319
If you restrict the
domain in the right way,

00:01:56.319 --> 00:02:02.356
inverse tangent of the
tangent of something,

00:02:02.356 --> 00:02:06.705
let's say theta is just
going to be equal to theta.

00:02:06.705 --> 00:02:08.914
Once again if we restrict the domain,

00:02:08.914 --> 00:02:11.262
if we restrict what the possible values

00:02:11.262 --> 00:02:13.224
of theta are in the right way.

00:02:13.224 --> 00:02:15.935
Let's just assume that we're doing that

00:02:15.935 --> 00:02:19.309
and so the inverse
tangent of the tan, of this

00:02:19.309 --> 00:02:21.694
is going to be just this
stuff right over here.

00:02:21.694 --> 00:02:23.305
It's just going to be that,

00:02:23.305 --> 00:02:26.479
it's going to be g inverse of x

00:02:26.479 --> 00:02:29.303
minus three pi over two.

00:02:29.303 --> 00:02:30.889
Now we're in the [home] stretch

00:02:30.889 --> 00:02:32.142
to solve for g inverse of x

00:02:32.142 --> 00:02:34.496
we could just add three
pi over two to both sides

00:02:34.496 --> 00:02:37.668
so we get and actually let
me just swap both sides.

00:02:37.668 --> 00:02:42.255
We get g inverse of x is equal to

00:02:42.255 --> 00:02:47.386
the inverse tangent of x minus six

00:02:47.386 --> 00:02:49.751
and then we're adding three
pi over two to both sides

00:02:49.751 --> 00:02:52.235
so this side is now on this side

00:02:52.235 --> 00:02:58.806
so plus three pi over two.

00:02:58.806 --> 00:03:00.585
Let me actually type that

00:03:00.585 --> 00:03:02.085
and I'm going to see if I can remember it

00:03:02.085 --> 00:03:03.958
because I'm about to
lose this on my screen

00:03:03.958 --> 00:03:08.117
so inverse tangent of x minus
six plus three pi over two

00:03:08.117 --> 00:03:09.493
so let me write that down.

00:03:09.493 --> 00:03:10.760
Let me type this.

00:03:10.760 --> 00:03:16.168
G inverse of x is going
to be the inverse tangent

00:03:16.168 --> 00:03:17.681
so I can write it like this,

00:03:17.681 --> 00:03:21.237
the inverse tangent of x minus six

00:03:21.237 --> 00:03:23.169
and yes it interpreted it correctly.

00:03:23.169 --> 00:03:24.684
Inverse tangent you can do that as

00:03:24.684 --> 00:03:30.599
arctangent of x minus six
plus three pi over two

00:03:30.599 --> 00:03:32.930
and it did interpret it correctly

00:03:32.930 --> 00:03:34.506
but then we have to think about

00:03:34.506 --> 00:03:37.106
what is the domain of g inverse?

00:03:37.106 --> 00:03:39.951
What is the domain of g inverse of x?

00:03:39.951 --> 00:03:41.696
Let's think about this a little bit more.

00:03:41.696 --> 00:03:45.123
The domain of g inverse of x,

00:03:45.123 --> 00:03:48.534
so let's just think about
what tangent is doing.

00:03:48.534 --> 00:03:54.269
The tangent function if
we imagine a unit circle,

00:03:54.269 --> 00:03:56.267
so that's a unit circle right over there.

00:03:56.267 --> 00:03:58.615
Guess we can imagine to be a unit circle.

00:03:58.615 --> 00:04:00.889
My pen tool is acting up a little bit

00:04:00.889 --> 00:04:02.522
it's putting this little gaps and things

00:04:02.522 --> 00:04:04.844
but I think we can power through that.

00:04:04.844 --> 00:04:07.041
Let's just say for the sake of argument

00:04:07.041 --> 00:04:08.217
that that's a unit circle,

00:04:08.217 --> 00:04:10.173
that's the x axis and that's the y axis.

00:04:10.173 --> 00:04:12.359
If you form an angle theta.

00:04:12.359 --> 00:04:15.421
If you form some angle
theta right over here,

00:04:15.421 --> 00:04:19.082
the tangent of theta is
essentially the slope

00:04:19.082 --> 00:04:22.538
of this terminal ray
of the angle or the ...

00:04:22.538 --> 00:04:24.628
Or I guess we can call it the
terminal ray of the angle.

00:04:24.628 --> 00:04:25.915
The angles form by that ray

00:04:25.915 --> 00:04:28.342
and this ray along the positive x axis.

00:04:28.342 --> 00:04:34.713
The tangent of theta is
the slope right over there

00:04:34.713 --> 00:04:39.214
and you can get a tangent of
any theta except for a few.

00:04:39.214 --> 00:04:41.173
You can find the tangent of that,

00:04:41.173 --> 00:04:42.335
you could find the slope there,

00:04:42.335 --> 00:04:43.403
you could find the slope there,

00:04:43.403 --> 00:04:45.324
you could also find the slope there,

00:04:45.324 --> 00:04:48.408
you could find the slope there

00:04:48.408 --> 00:04:50.461
but the place where you
can't find the slope

00:04:50.461 --> 00:04:55.521
is when this ray goes straight up,

00:04:55.521 --> 00:04:57.778
or this ray goes straight down.

00:04:57.778 --> 00:05:00.244
Those were the cases where
you can't find the slope.

00:05:00.244 --> 00:05:01.917
They are the slope you
could say is approaching

00:05:01.917 --> 00:05:03.968
positive or negative infinity.

00:05:03.968 --> 00:05:10.511
The domain of tangent, so tangent domain

00:05:10.511 --> 00:05:15.685
so the domain is essentially
all real numbers,

00:05:15.685 --> 00:05:26.085
all reals except multiples of pi over …

00:05:26.085 --> 00:05:29.867
I guess you can say pi over
two plus multiples of pi,

00:05:29.867 --> 00:05:35.303
except pi over two plus multiples of pi

00:05:35.303 --> 00:05:37.020
where k could be any integer

00:05:37.020 --> 00:05:38.442
so you could also be subtracting pi

00:05:38.442 --> 00:05:39.971
because if you have pi over two,

00:05:39.971 --> 00:05:41.517
if you add pi, you go straight down here.

00:05:41.517 --> 00:05:42.627
You add another pi you go up there,

00:05:42.627 --> 00:05:44.074
if you subtract pi you go down here,

00:05:44.074 --> 00:05:46.603
add, subtract another
pi you go over there.

00:05:46.603 --> 00:05:48.277
This is the domain

00:05:48.277 --> 00:05:52.714
but given this domain you
can get any real number.

00:05:52.714 --> 00:06:00.119
The range here is all reals

00:06:00.119 --> 00:06:02.647
because you can get any slope here,

00:06:02.647 --> 00:06:05.572
you can increase theta if
you want a really high slope,

00:06:05.572 --> 00:06:07.721
decrease theta if you want
a really negative slope

00:06:07.721 --> 00:06:08.675
right over there.

00:06:08.675 --> 00:06:11.204
You can really get to anything.

00:06:11.204 --> 00:06:16.866
Now when you're taking
about the inverse tangent,

00:06:17.081 --> 00:06:18.742
by convention you're going to …

00:06:18.742 --> 00:06:21.315
Well to make tangent invertible

00:06:21.315 --> 00:06:23.169
so that you don't have multiple elements

00:06:23.169 --> 00:06:25.257
of your domain all mapping
to the same element

00:06:25.257 --> 00:06:27.029
of the range because for example,

00:06:27.029 --> 00:06:30.980
this angle right over here
has the exact same slope

00:06:30.980 --> 00:06:34.489
as this angle right over here.

00:06:37.073 --> 00:06:40.785
If you have two theta's
mapping to the same tangent

00:06:40.785 --> 00:06:43.213
then that's not, if you
don't restrict your domain

00:06:43.213 --> 00:06:44.561
so that you only have one of them,

00:06:44.561 --> 00:06:46.112
it's not going to be invertible

00:06:46.112 --> 00:06:52.429
so the convention is is that
to make tangent invertible

00:06:52.429 --> 00:07:00.194
you restrict its domain

00:07:01.086 --> 00:07:05.237
to the interval from negative pi over two

00:07:05.237 --> 00:07:09.877
to pi over two in order to
construct the inverse tangent.

00:07:09.877 --> 00:07:12.397
The inverse tangent, you can input

00:07:12.397 --> 00:07:14.493
any real number into it

00:07:14.493 --> 00:07:16.518
so the inverse tangent's domain,

00:07:16.518 --> 00:07:19.272
this is just the convention.

00:07:19.272 --> 00:07:21.910
They could have restricted
tangents domains

00:07:21.910 --> 00:07:25.882
as long as for any theta,
there's only one theta

00:07:25.882 --> 00:07:29.593
in that domain that maps to a
specific element of the range

00:07:29.593 --> 00:07:33.500
but the convention is,
well inverse tangent can …

00:07:33.500 --> 00:07:36.486
The convention is to
restrict tangents domain

00:07:36.486 --> 00:07:38.782
between negative pi over
two and pi over two.

00:07:38.782 --> 00:07:44.329
Inverse tangents domain is all reals

00:07:44.329 --> 00:07:46.575
but its range is restricted.

00:07:46.575 --> 00:07:49.311
Its range and this is by convention

00:07:49.311 --> 00:07:52.109
it's going to be between
negative pi over two

00:07:52.109 --> 00:07:56.536
and pi over two and not including them.

00:07:56.536 --> 00:07:59.497
Le'’s go back to our original
question right over here,

00:07:59.497 --> 00:08:02.322
what is the domain of g inverse?

00:08:02.322 --> 00:08:06.522
Let's look at our domain of g inverse,

00:08:06.522 --> 00:08:08.278
well g inverse the domain of this

00:08:08.278 --> 00:08:14.643
I could put any real number in here.

00:08:14.643 --> 00:08:16.472
Now what this is going to pop out

00:08:16.472 --> 00:08:17.849
is going to be something

00:08:17.849 --> 00:08:19.919
between negative pi
over two and pi over two

00:08:19.919 --> 00:08:21.917
but they're not asking us
the range of g inverse.

00:08:21.917 --> 00:08:24.274
Actually would have been a
more interesting question.

00:08:24.274 --> 00:08:26.812
They're asking us what's
the domain of g inverse

00:08:26.812 --> 00:08:29.871
and I could put in any real
number right here for x,

00:08:29.871 --> 00:08:32.381
so let's put that in here.

00:08:32.381 --> 00:08:33.982
Domain of g inverse of x,

00:08:33.982 --> 00:08:36.629
it's negative infinity to infinity

00:08:36.629 --> 00:08:37.808
but actually just for fun

00:08:37.808 --> 00:08:39.598
and let's just verify that
we got the question right

00:08:39.598 --> 00:08:40.426
and we did

00:08:40.426 --> 00:08:41.525
but just for fun.

00:08:41.525 --> 00:08:42.681
Actually I am curious,

00:08:42.681 --> 00:08:45.266
let's think about what
the range of g inverse is.

00:08:45.266 --> 00:08:47.814
The range of this thing right over here

00:08:47.814 --> 00:08:51.431
is going to be between
negative pi over two

00:08:51.431 --> 00:08:55.542
to pi over two, that's for
this part right over here

00:08:55.542 --> 00:08:59.105
and then we're going to add
three pi over two's to it.

00:08:59.105 --> 00:09:02.849
The range for the entire function,

00:09:02.849 --> 00:09:08.654
so the range for this
thing is going to be,

00:09:08.654 --> 00:09:12.218
what the low end if we add
three pi over two to this,

00:09:12.218 --> 00:09:14.683
this is going to give us two pi over two

00:09:14.683 --> 00:09:16.005
which is just going to be,

00:09:16.005 --> 00:09:18.013
so three pi over two minus pi over two

00:09:18.013 --> 00:09:20.604
is going to be two pi
over two which is just pi,

00:09:20.604 --> 00:09:23.555
just pi all the way to three pi over two

00:09:23.555 --> 00:09:26.030
plus another pi over two is
going to be four pi’s over two

00:09:26.030 --> 00:09:28.256
or two pi.

00:09:28.256 --> 00:09:32.306
The range of g inverse
of x is pi to two pi

00:09:32.306 --> 00:09:34.410
and it's an open interval.

00:09:34.410 --> 00:09:36.076
Doesn't include the boundaries

00:09:36.076 --> 00:09:40.119
but its domain you could
put any value for x here

00:09:40.119 --> 00:09:42.351
and it will be defined.


Restricting domain of trig function to make invertible

WEBVTT
Kind: captions
Language: en

00:00:00.774 --> 00:00:02.508
To which intervals could we restrict

00:00:02.508 --> 00:00:04.977
f of x is equal to cosine of x minus pi

00:00:04.977 --> 00:00:08.911
over four, so that f of x is invertible?

00:00:08.911 --> 00:00:12.177
And they show us what cosine
of x minus pi over four,

00:00:12.177 --> 00:00:14.367
what it looks like, the graph of it.

00:00:14.367 --> 00:00:15.978
So lets just think about what it means

00:00:15.978 --> 00:00:17.642
for a function to be invertible.

00:00:17.642 --> 00:00:19.774
A function is a mapping from a set of

00:00:19.774 --> 00:00:22.136
elements that we would call the domain,

00:00:22.910 --> 00:00:24.712
so let me, my pen is a little

00:00:24.712 --> 00:00:26.838
off today so lets see if it works ok.

00:00:26.838 --> 00:00:30.034
So this right here is our domain.

00:00:30.034 --> 00:00:36.434
And this over here is our range.

00:00:36.434 --> 00:00:39.510
And a function maps from
an element in our domain,

00:00:39.510 --> 00:00:41.311
to an element in our range.

00:00:41.311 --> 00:00:43.105
That's what a function does.

00:00:43.105 --> 00:00:45.437
Now the inverse of the
function maps from that

00:00:45.437 --> 00:00:49.640
element in the range to
the element in the domain.

00:00:49.640 --> 00:00:51.637
So that over there would be f inverse.

00:00:51.637 --> 00:00:53.377
If that's the direction of the function,

00:00:53.377 --> 00:00:55.771
that's the direction of f inverse.

00:00:55.771 --> 00:00:58.636
Now, a function is not invertible,

00:00:58.636 --> 00:01:01.573
one of the situations in which
a function is not invertible

00:01:01.573 --> 00:01:03.914
you could have a function
where two elements of the

00:01:03.914 --> 00:01:07.546
domain map to the same
element of the range.

00:01:08.112 --> 00:01:11.568
So both of these elements map
to that element of the range,

00:01:11.568 --> 00:01:13.644
so both of these are the function,

00:01:13.644 --> 00:01:16.711
but then, if this is the
case, you're not going

00:01:16.711 --> 00:01:18.629
to be able to create a function that maps

00:01:18.629 --> 00:01:20.301
the other way, because if you input this

00:01:20.301 --> 00:01:23.045
into the inverse
function, where do you go?

00:01:23.045 --> 00:01:24.646
Do you go to that element of the domain?

00:01:24.646 --> 00:01:27.100
Or do you go to that
element right over there?

00:01:27.100 --> 00:01:28.571
So one way to think about it is,

00:01:28.571 --> 00:01:30.443
you need a one to one mapping.

00:01:30.443 --> 00:01:34.508
For each element of the
range, there's only one

00:01:34.508 --> 00:01:37.444
element of the domain that gets you there.

00:01:37.444 --> 00:01:38.972
Or another way to think
about it, you could try to

00:01:38.972 --> 00:01:41.908
draw a horizontal line on
the graph of the function

00:01:41.908 --> 00:01:45.914
and see if it crosses through
the function more than once.

00:01:45.914 --> 00:01:47.447
And you could see that this is indeed the

00:01:47.447 --> 00:01:49.573
case for this function right over here.

00:01:49.573 --> 00:01:52.027
If I did a horizontal
line, right over here.

00:01:52.027 --> 00:01:53.776
Now why is this the issue?

00:01:53.776 --> 00:01:55.576
Well this is showing, actually let me show

00:01:55.576 --> 00:01:59.245
a number that's a little
bit easier to look at.

00:01:59.245 --> 00:02:01.567
So lets say I drew the
horizontal line right over here.

00:02:01.567 --> 00:02:04.177
Now why is this horizontal line an issue?

00:02:04.177 --> 00:02:07.779
Well, showing us the
part of the domain that's

00:02:07.779 --> 00:02:10.113
being graphed here, that
there's several points

00:02:10.113 --> 00:02:12.577
that map to the same element of the range.

00:02:12.577 --> 00:02:15.639
They're mapping to 0.5.

00:02:15.639 --> 00:02:19.376
0.5, this value right over here.

00:02:19.376 --> 00:02:23.439
When you take f of that
is equal to 0.5, f of this

00:02:23.439 --> 00:02:29.635
is equal to 0.5, f of this
right over here is 0.5.

00:02:29.635 --> 00:02:32.704
So if you have multiple
elements of your domain

00:02:32.704 --> 00:02:34.435
mapping to the same element
of the range, then the

00:02:34.435 --> 00:02:37.837
function will not be
invertible for that domain.

00:02:37.837 --> 00:02:39.170
So really what we're going to do is,

00:02:39.170 --> 00:02:41.498
we're going to try to
restrict the domain so that,

00:02:41.498 --> 00:02:44.509
for that domain, if I were
to essentially apply this,

00:02:44.509 --> 00:02:46.646
what I call the horizontal line test,

00:02:46.646 --> 00:02:49.246
I'd only intersect the function once.

00:02:49.246 --> 00:02:52.169
So lets look at the graph
of the function once.

00:02:52.169 --> 00:02:54.166
So lets look at these choices.

00:02:54.166 --> 00:02:59.633
So the first one is an open
set from -5 pi over four.

00:02:59.633 --> 00:03:02.511
-5 pi over four, so that's pi, that's

00:03:02.511 --> 00:03:05.713
negative pi, and another fourth of pi.

00:03:05.713 --> 00:03:08.700
So that's, I think,
starting right over here,

00:03:08.700 --> 00:03:11.773
going all the way to negative 1/4th pi.

00:03:11.773 --> 00:03:14.509
So that's this domain right over here.

00:03:14.509 --> 00:03:16.570
Let me do this in a new color.

00:03:16.570 --> 00:03:23.401
So that's this, and this does
not include the two endpoints.

00:03:23.444 --> 00:03:26.578
So here I can still apply
the horizontal line and,

00:03:26.578 --> 00:03:30.311
in that domain, I can show
that there's two members

00:03:30.311 --> 00:03:33.973
of the domain that are
mapping to the same element

00:03:33.973 --> 00:03:37.113
in the range, and so if
I'm trying to construct

00:03:37.113 --> 00:03:40.646
the inverse of that, what
would this element, which I

00:03:40.646 --> 00:03:44.514
guess is -0.6, what would
that f inverse of -0.6 be?

00:03:44.514 --> 00:03:46.301
Would it be this value here?

00:03:46.301 --> 00:03:48.775
Or would it be this value here?

00:03:48.775 --> 00:03:51.177
I would rule this one out.

00:03:51.177 --> 00:03:54.980
Let's see, negative pi to pi.

00:03:54.980 --> 00:03:59.233
I'll do this in this
color right over here.

00:03:59.233 --> 00:04:01.311
Negative pi to pi.

00:04:01.311 --> 00:04:04.375
This is a fairly, so once
again, this is enclosed,

00:04:04.375 --> 00:04:05.702
so we're including the two boundaries,

00:04:05.702 --> 00:04:10.902
we're including negative
pi and pi in the domain.

00:04:10.902 --> 00:04:12.979
But once again over that interval,

00:04:12.979 --> 00:04:17.038
I could apply my horizontal
line here and notice,

00:04:17.038 --> 00:04:18.177
or actually I could even
apply the original one

00:04:18.177 --> 00:04:21.437
that I did in blue, and
notice there's multiple

00:04:21.437 --> 00:04:25.839
elements of the domain
that map to, say 0.5.

00:04:25.839 --> 00:04:28.369
So what would f inverse of 0.5 be?

00:04:28.369 --> 00:04:30.573
You can't construct a function where it

00:04:30.573 --> 00:04:32.979
maps only to one element of the domain, so

00:04:32.979 --> 00:04:35.977
we could rule this one out right as well.

00:04:35.977 --> 00:04:40.577
Now, negative 1/2 pi to positive 1/2 pi.

00:04:40.577 --> 00:04:46.308
So negative 1/2 pi, so let
me, I'm running out of colors.

00:04:46.308 --> 00:04:51.745
So negative 1/2 pi to positive 1/2 pi,

00:04:52.163 --> 00:04:55.039
This one is interesting, if
I apply a horizontal line

00:04:55.039 --> 00:04:59.837
there, there, there, so
lets see, but if I apply

00:04:59.837 --> 00:05:02.977
a horizontal line right
over here I do intersect

00:05:02.977 --> 00:05:06.378
the function twice, so
I have two members of

00:05:06.378 --> 00:05:08.847
this domain mapping to the same element

00:05:08.847 --> 00:05:11.605
of the range, so I can
rule that one out as well.

00:05:11.707 --> 00:05:13.376
And I'm left with one last choice.

00:05:13.376 --> 00:05:15.300
I'm hoping this one will work out.

00:05:15.300 --> 00:05:19.512
So 1/2 pi, it's an open set, so 1/2 pi,

00:05:19.512 --> 00:05:23.380
right over there, to five pi over four.

00:05:24.764 --> 00:05:28.502
So that's pi and another 1/4th,
so that's right over there.

00:05:28.502 --> 00:05:32.890
And lets see, this is, if I
were to look at the graph here,

00:05:33.074 --> 00:05:35.445
it seems like it would pass
the horizontal line test.

00:05:35.445 --> 00:05:37.238
At any point here I could make

00:05:37.238 --> 00:05:38.976
a horizontal line over that domain.

00:05:38.976 --> 00:05:40.911
Actually, let me do it
for the whole domain.

00:05:40.911 --> 00:05:42.116
So you see that.

00:05:42.116 --> 00:05:43.378
For the whole domain.

00:05:43.378 --> 00:05:45.844
And I'm only intersecting
the function once.

00:05:45.844 --> 00:05:50.974
So, for every element of the range that

00:05:50.974 --> 00:05:52.777
we're mapping to, there's only one element

00:05:52.777 --> 00:05:54.644
in our domain that is mapping to it.

00:05:54.644 --> 00:05:56.910
It's passing our horizontal line test,

00:05:56.910 --> 00:05:59.337
so I would check this
one right over there.


Inverse tangent scenario

WEBVTT
Kind: captions
Language: en

00:00:00.983 --> 00:00:03.244
Javier is calibrating sophisticated

00:00:03.244 --> 00:00:05.388
medical imaging equipment.

00:00:05.388 --> 00:00:07.721
The manual reports that the tangent

00:00:07.721 --> 00:00:10.992
of a particular angle is one.

00:00:10.992 --> 00:00:12.896
So that's saying that the tangent

00:00:12.896 --> 00:00:14.240
and let's say that that particular angle

00:00:14.240 --> 00:00:16.977
is theta is equal to one.

00:00:17.210 --> 00:00:19.986
What should Javier do to find the angle?

00:00:19.986 --> 00:00:21.472
And I encourage you to pause this video

00:00:21.472 --> 00:00:22.864
and look at these choices

00:00:22.864 --> 00:00:23.837
and think about which of these

00:00:23.837 --> 00:00:27.829
should he do to find the angle?

00:00:28.898 --> 00:00:31.204
So let's look through each of them.

00:00:31.316 --> 00:00:32.211
So the first one...

00:00:32.211 --> 00:00:33.492
well, actually, instead
of looking at the choices,

00:00:33.492 --> 00:00:34.605
let's think about what we would do

00:00:34.605 --> 00:00:35.752
to find the angle.

00:00:35.752 --> 00:00:36.994
So they're saying that the tangent of

00:00:36.994 --> 00:00:39.979
some angle is equal to one.

00:00:39.979 --> 00:00:41.900
Well one thing that you
might want to do is say

00:00:41.900 --> 00:00:45.137
okay, if we take the inverse tangent,

00:00:45.137 --> 00:00:47.121
if we take the inverse tangent,

00:00:47.121 --> 00:00:50.273
of the tangent of theta,

00:00:50.273 --> 00:00:51.792
so if we take the inverse tangent

00:00:51.792 --> 00:00:53.221
of both sides of this,

00:00:53.221 --> 00:00:54.189
we of course would get

00:00:54.189 --> 00:00:56.564
the inverse tangent of
the tangent of theta.

00:00:56.564 --> 00:01:00.338
If the domain over here is
restricted appropriately,

00:01:00.338 --> 00:01:02.390
is just going to be equal to theta,

00:01:02.390 --> 00:01:05.698
so we could say the theta

00:01:05.698 --> 00:01:07.147
is going to be the inverse tangent of one.

00:01:07.147 --> 00:01:08.035
So it might be tempting to just

00:01:08.035 --> 00:01:09.908
pick this one right over here.

00:01:09.908 --> 00:01:12.436
Type inverse tangent of
one into his calculator.

00:01:12.436 --> 00:01:16.787
So maybe this looks like the best choice.

00:01:16.874 --> 00:01:17.696
But remember,

00:01:17.696 --> 00:01:21.217
I said if we restrict the
domain right over here.

00:01:21.217 --> 00:01:23.086
if we restrict the
possible values of tangent,

00:01:23.086 --> 00:01:25.113
of theta here appropriately,

00:01:25.113 --> 00:01:29.151
then this is going to simplify to this.

00:01:29.151 --> 00:01:30.190
But there is a scenario

00:01:30.190 --> 00:01:31.852
where this does not happen.

00:01:31.852 --> 00:01:35.191
And that's if we pick
thetas that are outside

00:01:35.191 --> 00:01:38.698
of the range of the
inverse tangent function.

00:01:38.698 --> 00:01:40.074
What do I mean by that?

00:01:40.074 --> 00:01:42.006
Well it's really just based on the idea

00:01:42.006 --> 00:01:45.262
that there are multiple
angles that have...

00:01:45.262 --> 00:01:48.930
or multiple angles whose tangent is one.

00:01:48.930 --> 00:01:49.870
And let me draw that here

00:01:49.870 --> 00:01:51.332
with a unit circle here.

00:01:51.332 --> 00:01:53.799
So we draw a unit circle,

00:01:54.416 --> 00:01:56.242
so that's my x axis,

00:01:56.242 --> 00:01:58.304
that's my y axis,

00:01:58.304 --> 00:02:01.306
let me draw my unit circle here.

00:02:01.752 --> 00:02:02.603
Actually you probably don't even

00:02:02.603 --> 00:02:03.756
have to draw the unit circle,

00:02:03.756 --> 00:02:05.097
because the tangent is really much more

00:02:05.097 --> 00:02:07.278
about the slope of the ray

00:02:07.278 --> 00:02:08.573
created by the angle,

00:02:08.573 --> 00:02:10.532
than where it intersects the unit circle

00:02:10.532 --> 00:02:13.035
as would be the case with sine and cosine.

00:02:13.035 --> 00:02:14.356
So if you have..

00:02:14.356 --> 00:02:17.014
so you could have this
angle right over here.

00:02:17.418 --> 00:02:20.477
So let's say this is a candidate theta,

00:02:20.477 --> 00:02:22.016
where the tangent of this theta is

00:02:22.016 --> 00:02:23.891
the slope of this line,

00:02:23.891 --> 00:02:25.903
and this terminal angle,

00:02:25.903 --> 00:02:27.086
the terminal ray, you could say

00:02:27.086 --> 00:02:28.152
of the angle.

00:02:28.152 --> 00:02:29.914
The other side, the initial ray,

00:02:29.914 --> 00:02:31.731
is along the positive x axis.

00:02:31.924 --> 00:02:32.738
And so you could say,

00:02:32.738 --> 00:02:34.632
okay the tangent of this theta,

00:02:34.632 --> 00:02:36.065
the tangent of this theta is one.

00:02:36.065 --> 00:02:38.148
Because the slope of this line is one.

00:02:38.148 --> 00:02:39.056
Let me scroll over a little bit.

00:02:39.056 --> 00:02:40.685
Well, so let me write it this way.

00:02:40.685 --> 00:02:44.401
So tangent theta is equal to one.

00:02:44.550 --> 00:02:46.170
But I can construct another theta

00:02:46.170 --> 00:02:48.052
whose tangent is equal to one

00:02:48.052 --> 00:02:49.533
by going all the way over here

00:02:49.533 --> 00:02:51.704
and essentially going in
the opposite direction

00:02:51.704 --> 00:02:53.640
but the slope of this line,

00:02:53.640 --> 00:02:55.478
so let's call this theta two,

00:02:55.478 --> 00:02:59.110
tangent of theta two is also

00:02:59.110 --> 00:03:00.569
going to be equal to one.

00:03:00.569 --> 00:03:02.266
And of course you could go another

00:03:02.266 --> 00:03:04.649
pi radiance and go back
to the original angle,

00:03:04.649 --> 00:03:07.991
but that's functionally the same angle

00:03:07.991 --> 00:03:10.822
in terms of where it is relative

00:03:10.822 --> 00:03:12.560
to the positive x axis,

00:03:12.560 --> 00:03:13.993
or what direction it points into,

00:03:13.993 --> 00:03:15.196
but this one is fundamentally

00:03:15.196 --> 00:03:16.818
a different angle.

00:03:16.818 --> 00:03:18.647
So we do not know,

00:03:18.647 --> 00:03:20.858
we do not have enough information

00:03:20.858 --> 00:03:22.718
just given what we've been told

00:03:22.718 --> 00:03:23.806
to know exactly

00:03:23.806 --> 00:03:25.113
which theta we're talking about,

00:03:25.113 --> 00:03:26.846
whether we're talking
about this orange theta

00:03:26.846 --> 00:03:28.336
or this mauve theta.

00:03:28.336 --> 00:03:30.436
So I would say the get more information,

00:03:30.436 --> 00:03:34.329
there are multiple angles
which fit this description.