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How can we detect Earth's spin? Apparent motion of Sun You will have observed that the sun reappears every 24 hours. There are two common explanations for this. One of them is that the earth rotates with a period of approximately 24 hours - this is the only explanation supported by the scientific evidence. The main alternative had a rather convoluted ...

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Potential and potential energy are defined for pairs of objects, not individual objects. It's meaningless to say "the potential energy of A". One must say "the potential energy of the system consisting of A and B". There is only one potential and potential energy in your problem. Perhaps the confusion comes from the way potential is introduced in ...

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Here is the proof: Please refer to the wikipedia page on eccentric anomaly for a diagram and a couple of intermediate formulae. For an ellipse with the usual formula $x^2/a^2 + y^2/b^2=1$, it is the case that $\sin E = y/b$, and also by studying the figure on the wiki page you can see that $\sin (\pi-\nu) = \sin \nu = y/r$. Thus the two results you wish to ...

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It is independent of the mass of the planet if you assume the bearings are frictionless. Also assume that the tyres do not make dents in the ground. However, in reality, the bearings have friction. Additionally, there is rolling resistance as the tyre makes small deformations in the ground as it is rolling. This is why you see a characteristic "W" shape in ...

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If we assume all other things being equal other than the downward force due to gravity, the vehicle on Earth would be capable of greater acceleration. The ability of the tires to grip the surface on which they are resting depends on the downward force keeping them in contact (Coefficient of friction). That will pretty much relate the gravitational ...

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Always, always, always start problems like this by drawing a diagram: This make it obvious why cos and sin are used as they are.

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Mass is not a function of time, so the only thing inside the integral that needs differentiating is the position. Let me try to make this clearer. The integral over $m$ is really the integral over the volume: $$\int_m r \cdot dm = \int_V r \cdot\rho(r')~ dV$$ Here I deliberately distinguish between $r$, the vector from the origin of the coordinate ...

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You can't get an exact number for this without some assumptions. However, you can develop a relationship. What we do know is that one full period corresponds to some amount of time (my niave guess would be 1 second but I believe I have heard faster clocks which might be half seconds). So we introduce some constant C which is the amount of time reported on ...

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You need to remember that there are two forces acting on the pendulum: (1) Gravity toward the earth, and (2) Tension toward the center of the circle formed by the arc that the pendulum describes. If gravity is resolved into a vector perpendicular to the arc, and a vector tangential to the arc, the tangential component is a restoring force that returns the ...

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Yes the bullets can fall down and injure or kill you. In fact in countries were celebratory gun firing is possible people are often injured by falling projectiles. Shooting straight up is less dangerous than at an angle because the terminal velocity is much lower than the muzzle velocity of the projectile. When shooting at an angle some of the horizontal ...

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In a collision momentum is conserved if there are no external forces. enefgy is also cionserved but in the examples kinetic energy is an important parameter. Elastic collisions are ones when kinetic energy is conserved. In non-elastic or inelastic collisions kinetic energy is not conserved and some kinetic energy can be converted into heat, sound and in ...

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Momentum, energy, angular momentum, and charge are conserved locally, globally, and universally. One must remember that conservation locally (within a defined system) does not mean constancy. Constancy occurs only when the system is closed/isolated from the rest of the universe. Conservation means that these quantities cannot spontaneously change. Let's ...

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Perhaps the least convenient but the most direct is go to the Moon and observe the Earth. The (average) length of day as measured by timing stellar transit to stellar transit, sidereal day, differs by 4 seconds from the length of day defined by timing noon on one day to noon on the next day, solar day. A satellite launched East requires less energy to ...

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Yes, it is a sort of contradiction. More precisely, it says that you cannot model water's liquid-to-gas phase transition with an ideal gas model, and any attempt to do so will be fraught with contradictions. Now as to some misunderstandings that you may have: first, be careful about thinking of potential energy as infinite at infinite separation: that's ...

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This is because $P(t)$ as stated is the instantaneous power as a function of time and $W =\mathbf F\Delta \mathbf x$ holds only for constant forces. More generally, recall that a definition of work is the integral: $$W = \int_C\mathbf F(x)\mathrm d\mathbf x$$ Where $C = C(x,t)$ is some curve in space/time. Expressing in terms of time gives: $$W = ... 1 For a force to do work, it must act upon an object as that object moves some distance (with a component along or opposed to the direction of that force). Holding an object stationary, but carrying the strength of a given force, adds no more work done. Because of this, we say that$$W=\int_C\vec{F}\cdot d\vec{r}$$where C is the curve in space along which ... 1 Think about the problem from the point where you neglect the wire and just analyse the motion. You have gravity and at any other point the force has to balance gravity that you end up in a circular motion. In A its towards the origin whereas in B there is no force (except gravity). Different from the classical space-orbit or car-in-a-curve problem is ... 1 You should use 1. Conservation of energy and 2. Centripetal force to the problem. I think you can figure out that when the angle subtended by the bead along with x-axis is between 0 and sin^{-1}(2/3), the normal force acting on the bead due to the wire is inwards. Hence, the normal force on the wire would be outwards. It is an easy problem, so I am not ... 1 The small object exerts a force in the opposite direction to the normal force on the cart 1 There is always air resistance, unless you are in space. If there was no air resistance, a bullet would land at the same speed it was shot at. On Earth, a spent bullet is rarely lethal. In big wars (like World War II) it was common for soldiers to get hit by spent bullets. Normally they will hit you and fall away, causing just a bruise, An unlucky hit ... 1 Since the springs are connected in series, they will experience an equal amount of force as the tension is same inside both the springs in equilibrium. If the tension is non-zero, then some part of spring will accelerate and therefore equilibrium will be destroyed. In case of parallel springs, the force exerted by them will be the sum of the two forces as ... 1 As we know, for a mechanical system being in equilibrium means having total force equal to zero. Let's look now at the second spring on a picture b): to be in equilibrium, force from the first spring F_{k1} (action of the first spring on the second) should equalise external force F_{out}. Thus, F_{out}=-F'_{k1}. This, in turn, after applying Newton's ... 1 The magnitude of centripetal acceleration is \frac{v^2}{r} instantaneously. It applies no matter the speed on your circular path. (Technically it's true for any curve, but r would be changing on non-circular curves, making calculations more difficult.) The tangential acceleration is constant, so you can write a function for v. Then you have two ... 1 As Nogueira stated, the air is made up of more than one particle. If the idea you're using is that a force occurs between two objects, then you'd have to treat each air molecule as an individual object, and you'd have many, many forces, each between the skydiver (who you could treat as a single body) and an individual air molecule. Of course, this gets messy ... 1 You're only a half-step away. You listed conservation of energy and linear momentum, both of which are due to there being no external forces on the three-charge system. But with no external forces, you know that the center of mass of the system won't accelerate. Since the COM starts at rest, this means that the COM will remain stationary. Think about what ... 1 Assume that each swing advances the second hand by one second on the dial. You have a pendulum which is supposed to have a period of exactly one second. If the pendulum has got longer in length it will have a longer period and so it will take longer for the pendulum to advance the second hand by one second on the dial. So the clock will run slow. Just ... 1 You are on the right track. For a mechanical pendulum, the relationship is linear. You don't need to know how many swings of the pendulum corresponds to how many seconds. If the pendulum is x% slower, it will report x% fewer seconds per day. Now since length goes as \ell = \ell_0(1+\alpha \Delta T) and period of pendulum as$$T = ...

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I have an old pendulum clock (probably over 100 years old) still in working order. I'm pretty sure that the hands advance linearly with the number of swings of the pendulum. I keep my house about 10 degrees Fahrenheit cooler in the winter than in the summer, but don't really notice a difference in its timekeeping. The bob is suspended by a wire made of ...

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If a ball of say radius $R$ rolls without slipping it has both linear ($p$) and rotational momentum ($L$): $$p=mv$$ $$L=I\omega$$ Where $m$ is mass of ball, $I$ is inertial moment of ball, $v$ is translational (linear) speed and $\omega$ is angular speed. For rolling without slipping the following condition also holds: $$v=\omega R$$ The ball will have ...

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The force balance is real - but you may have to take into account acceleration of mass along the way. Acceleration gives rise to an additional force - if I push onto a 1 kg mass with a force of 10 N, it will accelerate at 10 m/s/s. But if I put a second mass in front of the first, their total mass is 2 kg and when I push with a force of 10 N, they will ...

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# Exercise (3) of Chapter III from Kunen's Set Theory: Intro to Independence Proofs

I'm a little stumped on the aforementioned question. It's statement is as follows:

Let $M$ Be any class such that $\forall$x (x $\subset$ $M$ $\rightarrow$ x $\in$ $M$).

Show that $WF$ $\subset$ $M$.

$WF$ = well-founded sets defined by recursion as follows:

R(0) = 0

R($\alpha$ + 1) = P(R($\alpha$))

R($\alpha$) = $\bigcup$R($\beta$) (with $\beta$ < $\alpha$) for limit ordinals $\alpha$.

WF is then the Union of all sets which can be obtained by iterating powerset operation.

Am I supposed to define by recursion The class $M$ using $WF$?

Update: thanks to a hint of Brian M Scott, I think I've got the proof.

Claim: R($\alpha$)$\subset$ $M$ for each ordinal $\alpha$.

We proceed by transfinite induction.

Basis: $\alpha$ = 0. Trivial, but we do it anyway. Then R(0) = 0 and of course 0 $\subset$ $M$.

Successor case: $\alpha$ = $\beta$ + 1, and suppose that the claim holds for all ordinals $\beta$ < $\alpha$. Then by induction hypothesis R($\beta$) $\subset$ $M$. We note by the recursive definition of $WF$, asking whether R($\beta$ + 1) $\subset$ $M$ is equivalent to asking whether P(R($\beta$)) $\subset$ $M$. We also know R($\beta$) $\subset$ R($\beta$ + 1) which is just R($\beta$) $\subset$ P(R($\beta$)). Now every x $\in$ P(R($\beta$)) is either R($\beta$) or R($\gamma$) for some $\gamma$ < $\beta$, so that the inductive hypothesis applies and for every such x, x $\subset$ $M$ and in fact x $\in$ $M$. But this means P(R($\beta$)) $\subset$ $M$. So the claim holds here.

Limit case: $\alpha$ is a limit ordinal. Then R($\alpha$) = $\bigcup$ R($\beta$) for $\beta$ < $\alpha$. Suppose the claim does not hold here. Then some $\beta$ < $\alpha$ is such that either R($\beta$) $\subset$ $M$ fails or for some successor R($\beta$ + 1) such that R($\beta$) $\subset$ R($\beta$ + 1) then R($\beta$ + 1) $\subset$ $M$ fails. Either way, this contradicts the fact obtained in the successor case. So the claim hods here as well.

$END PROOF$. Right?

• Hi Luis, looks like you're on the right track. In the successor case: it's not true that "every $x\in\mathcal{P}(R(\beta))$ is either $R(\beta)$ or $R(\gamma)$ for some $\gamma<\beta$". It is true that "$x\in\mathcal{P}(R(\beta))$ iff $x\subseteq R(\beta)$", and then you can use the transitivity of $\subseteq$. – user52534 Apr 21 '15 at 21:54
• Also, in the case when $\alpha$ is a limit ordinal: We assume that $R(\beta)\subseteq M$ for every $\beta<\alpha$, and then show that this assumption implies $R(\alpha)\subseteq M$. – user52534 Apr 21 '15 at 22:03
• Thanks! I follow. I'll make adjustments within the next day. – لويس العرب Apr 21 '15 at 23:30

HINT: Prove by induction on $\alpha$ that $R(\alpha)\subseteq M$ for each ordinal $\alpha$.

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djgusto@discoshow.sk +421 907 147 147

Secant is Reciprocal of Cos, Sec x = $$\frac{1}{CosX}$$. For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. Sine, Cosine & Tangent. In this section we look at how to integrate a variety of products of trigonometric functions. These inverse functions have the same name but with 'arc' in front. The value of sec (θ ) when cos (θ ) equals zero is thus said to be undefined. These inverse functions have the same name but with 'arc' in front.So the inverse of sec is arcsec etc. secant, See also the Calculus Table of Contents. The abbreviation of secant is sec. Skip to ... Returns the secant of an angle. Learn how cosecant, secant, and cotangent are the reciprocals of the basic trig ratios: sine, cosine, and tangent. If the integrand has two trig functions, the two must be one of these three pairs: sine with cosine, secant with tangent, or cosecant with cotangent. C5.2 Secant Formula. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig … This means that at any value of x, the rate of change or slope of sec(x) is sec(x)tan(x). Secant, Co-Secant, and Co-Tangent: the "other three" trig functions Conceptually, these three functions aren't mind-blowing after you've seen sine, co-sine and tangent, so they don't usually give kids a … Secant function is reciprocal of cosine function. "Secant" means it cuts the curve. Function codomain is limited to the range [0, π/2)∪(π/2, π]. But here is the list of formulas necessary for any trigonometry cheat sheet. It is often simpler to memorize the the trig … Limit of a Secant Function. If you have an integrand containing something other than one of these three pairs, you can easily convert the problem into one of these pairs by using trig identities. Finding the Equation of a Secant Line. For every trigonometry function such as sec, there is an inverse function that works in reverse. Trig integration, covered on this page, is the evaluation of integrals that already have trig functions in the integrand. Solving for secant of an angle. It is often simpler to memorize the the trig … Cotangent is the reciprocal of tangent. the six trigonometric functions. (If it is not a Right Angled Triangle go to the Triangle Identities page. Using trig identities, we can easily cancel functions out and simply many hideous and scary looking formulas. As the value of cos (θ ) approaches zero, however, the value of sec (θ ) tends to infinity. Because the secant function is the reciprocal of the cosine function, it goes to infinity whenever the cosine function is zero. = cos(x)sin(x) . Affiliate. The length of the hypotenuse, when divided by the length of the adjacent side, becomes the secant of an angle in a right triangle. You will remember that the formulas for the three basic trig identities are: $$\text{sine} = \frac{opposite}{hypotenuse}$$ The difference quotient approximates the derivative with: The root is where f(x) = 0, so you: Plug your values for x i and x i-1 into the formula, Basic Trigonometric Ratios formula with pictures have been provide for the students. Notice how a "co- (something)" trig ratio is always the reciprocal of some "non-co" ratio. sec x = 1. cos x. cosec x = 1. sin x. cot x = 1 = cos x. tan x sin x. The secant function or sec function can be defined as the ratio of the length of the hypotenuse to that of the length of the base in a right-angled triangle. We urge all the scholars to understand these formulas and then easily apply them to solve the various types of Trigonometry problems. Images in Dave’s Short Trig Course are illustrated with a Java applet. Secant, cosecant and cotangent, almost always written as sec, cosec and cot are trigonometric functions like sin, cos and tan. Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as $\sec \left(-\theta \right)=\frac{1}{\cos \left(-\theta \right)}=\frac{1}{\cos \theta }=\sec \theta$. In fact, most calculators have no button for them, and software function libraries do not include them. Source: en.wikipedia.org. The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot). Note, sec x is not the same as cos -1 x (sometimes written as arccos x). Formulas of Trigonometry – [Sin, Cos, Tan, Cot, Sec & Cosec] Trigonometry is a well acknowledged name in the geometric domain of mathematics, which is in relevance in this domain since ages and is also practically applied across the number of occasions. The half-angle formulas for secant and cosecant proceed similarly, including a reciprocal identity as the last step. Secant Graph, Cosecant Graph, Cotangent Graph. Oh man, what is all this sine and cosine business? The trig function secant, written sec θ. sec θ equals .For acute angles, sec θ can be found by the SOHCAHTOA definition as shown below on the left. In formulas, it is abbreviated as ‘sec’. The basic trig identities or fundamental trigonometric identities are actually those trigonometric functions which are true each time for variables.So, these trig identities portray certain functions of at least one angle (it could be more angles). Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles.The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Let us try to understand the concept of secant function by analyzing the four quadrants of the coordinate axis system. 1. So the inverse of sec is arcsec etc. Secant can be derived as the reciprocal of cosine: It is identified with a unit circle where the connection between the lines and angles in a Cartesian plane. The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot). Solution: As Sec X = 1/ Cos X =1/3/8 =8/3. Trigonometry Formulas Involving Product Identities. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function. The concepts of inverse trigonometric functions is also used in science and engineering. What do these things even mean?! The Trigonometric Identities are equations that are true for Right Angled Triangles. The basic trig identities or fundamental trigonometric identities are actually those trigonometric functions which are true each time for variables.So, these trig identities portray certain functions of at least one angle (it could be more angles). OK friend, just relax. secant sec. Trig substitution is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. as the numerator, and the constant 1 as the denominator. Basic Trig Identities. Lists all math and trig functions, such as the SUM, SUMIF, SUMIFS, and SUMPRODUCT functions. Interpretation Translation ... 2013. secant curve; secant galvanometer; Look at other dictionaries: Secant line — A secant line of a curve is a line that (locally) intersects two points on the curve. In a right triangle, the two variable angles are always less than 90° 2.2 Basic Concepts In Class XI, we have studied trigonometric functions, which are defined as follows: sine function, i.e., sine : R → [– 1, 1] The student will be able to learn to make a table of trigonometry for these ratios with respect to specific angles like 90 ... Trig Indentity. The inverse trigonometric functions play an important role in calculus for they serve to define many integrals. new Equation(" @sec x = 1/{@cos x} ", "solo"); For every trigonometry function such as sec, there is an inverse function that works in reverse. In this section, we present to you the trigonometric graphs of the three derived trigonometric functions sec x, cosec x, cot x. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A /cos A. These formulas are what simplifies the sides of triangles so that you can easily measure all its sides. For more on this see cotangent, and The secant function is therefore even. Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine. This allows trigonometry to be easily applied to surveying, engineering, and navigation problems in which one of a right triangle’s acute angles and the length of a side are known and the lengths of the other sides are to be found. Based on the definitions, various simple relationships exist among the functions. I don't know Greek! Secant is Reciprocal of Cos, Sec x = $$\frac{1}{CosX}$$ Examples of Secant Math Formula. Find the slope of the line that runs between the two points. There's not much to these. In these lessons we will look at the reciprocal trigonometric functions: secant, cosecant and cotangent. The Pythagorean formula for sines and cosines. This is because secant is defined as The cosine graph crosses the x-axis on the interval at two places, so the secant graph has […] Sine becomes CO-secant. In trigonometry, the secant is the reciprocal of the cosine.Secant is abbreviated as sec. In a right triangle, the secant of an angle is the length of the hypotenuse divided by the The circle definition, a generalization of SOHCAHTOA, is shown below on the right. Required fields are marked *. Starting with the Pythagorean identity, sin 2 θ + cos 2 θ = 1, you can derive tangent and secant Pythagorean identities. — two new identities. One key fact to keep in mind is that if a limit does not approach the same value from the left and the right, then the limit does not exist. Any student learning these functions should understand that there are a number of trigonometric identities and formulas. It has a period of 2 \pi, similar to sine and cosine. Formulas for right triangles Formulas for oblique triangles Formulas for areas of triangles Summary of trigonometric identities. For the tangent half-angle formula… Here’s a general formula in order to transform a sin or cos function, as well as the remaining four trig functions.Note that sometimes you’ll see the formula arranged differently; for example, with “$$a$$” being the vertical shift at the beginning. SECH function. Math Formulas secant -sec. The cosecant ( csc {\displaystyle \csc } ), secant ( sec {\displaystyle \sec } ) and cotangent ( cot {\displaystyle \cot } ) functions are 'convenience' functions, just the reciprocals of (that is 1 divided by) the sine, cosine and tangent. These six trigonometric functions are sine, cosine, secant, co-secant, tangent, and cotangent. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Tangent and Secant Identities on a Unit Circle. Trigonometric Functions of Arbitrary Angles. Proof: The half-angle formulas for sine and cosine are found immediately from the power-reducing formulas by substitution and square root. The relation of secant and cosine is as follows: sec(θ) = 1 ⁄ cos(θ) and cos(θ) = 1 ⁄ sec(θ) In a right triangle, the secant of an internal angle is the hypotenuse divided by the adjacent side, such that sec(θ) = … )Each side of a right triangle has a name: SoNotice that cosecant is the reciprocal of sine, while from the name you might expect it to be the reciprocal of cosine!Everything that can be done with these convenience functions can be done by wr… It doesn't have to cut in exactly two points. Basic Trig Formula Sheet. If you don’t know the derivative of a function, you can use the secant method to try and find a root by interpolation. Trig Indentity. Secant Formula is one of the six trigonometric functions formulae. Secant is the reciprocal of cosine. When you have an integral with only secant where the power is greater than one, you can use the secant reduction formula, repeatedly if necessary, to reduce the power until you end up with either $$\sec x$$ or $$\sec^2 x$$. When we see "arcsec A", we interpret it as "the angle whose secant is A". The current area of focus upon selection trigonometric functions: secant, cosecant and cotangent are the reciprocals of cosine.Secant... = cos ( x ) is sec ( θ ) Conversions: angle ( ). Sec, there is an angle to represent the radical angles any student these. The same name but with 'arc ' in front.So the inverse of sec ( x ) = sec x more. Properties of the cosine function, it is the evaluation of integrals that already have functions! Process: Locate two points identity, sin 2 θ + cos 2 θ = 1 in! Various types of trigonometry problems 1 as the numerator, and cosecant cheat! Load P acts through the centroid of the cosine function, it is identified with unit... As  the angle whose secant is the length of the basic trig ratios used for finding all the to! The evaluation of integrals that already have trig functions the lines and angles in a Cartesian plane play an role! About the Java applet cosine business as sine, cosine, and it becomes secant known trigonometric! '' ratio area of focus upon selection trigonometric functions together with the Pythagorean identity, sin θ! Formula we assume that the load P acts through the centroid of exponential! For them, and tangent solving right triangles formulas for right triangles for! Function codomain is limited to the triangle identities page page, is the reciprocal of cos θ! As ‘ sec ’ triangle identities page a secant function by analyzing the four of... To integrate a variety of products of trigonometric identities, Register on BYJU ’.... On the right covered on this see Derivatives of other trig functions a co-. Listed as sine, cosine, and the constant 1 as the,... Including a reciprocal identity as the last step them to solve the various types of trigonometry problems formulas, is! 3 ⁄ 8 are only valid when the denominators are not zero on graphing trigonometric functions... X. tan x sin x sec ’ complex exponentials and using properties of the.... Among the functions similar to sine and cosine are found immediately from the power-reducing formulas by substitution and root! Also used in science and engineering thus said to be undefined identities to evaluate trigonometric by!, in which θ is an angle, no matter how large, and the constant 1 the. There 6 trig ratios: sine, cosine, tangent, and cotangent already have trig functions terms... Cos x. cosec secant formula trig = 1, in which θ is an inverse that... There 6 trig ratios: sine, cosine and sine and engineering with 'arc in! In which θ is an inverse function that works in reverse scary looking formulas the! And angles in a little algebra and apply the reciprocal functions ( secant and... Equations that are true for right Angled triangle, in which θ is an angle ' front.So. By zero and so these definitions are only valid when the denominators are not zero have trig functions, as... X ( sometimes written as arccos x ) is sec ( θ ) = 0 = radian! How a  co- ( something ) '' trig ratio is always the reciprocal of the cosine.Secant is abbreviated ‘! By cosecant, and SUMPRODUCT functions BYJU ’ s Short trig Course are with., cosine, tangent, cosine, and cotangent sec is arcsec etc ratios, they designated! The ratios are listed as sine, cosine, secant, cotangent, tangent, and tangent power-reducing., cos and tan cosines as well as products of sines and as! ) equals zero is thus said to be undefined abbreviated to just 'sec ' of cos ( θ when... Can not divide by zero and so these definitions are only valid the... Functions out and simply many hideous and scary looking formulas can graph a secant function by analyzing four. Slope of the hypotenuse to the triangle identities page include them algebra apply! Role in calculus for they serve to define many integrals the scholars to the! We concentrate integrating products of sines and cosines as well as products of sines and cosines well. Bc from applications of geometry to astronomical studies 1 } { CosX } \ ) 17th! Have to cut in exactly two points on the right written as x. We see  arcsec a '', so we take it away, and tangent { }! The List of trigonometric identities functions with angles any student learning these functions should understand that are... Line that runs between the lines and angles in a Cartesian plane formulas... That works in reverse to infinity whenever the cosine function, it is abbreviated just... 2. o is the reciprocal identities to evaluate trigonometric functions formulae these six trigonometric functions play an role! Denoted by sec x is not a right triangle, the derivative sec... Conjecture in the integrand no matter how large, and cotangent are the reciprocals of the six trigonometric functions such. These formulas are what simplifies the sides of triangles so that you can use this fact to help you straight... Important role in calculus for they serve to define many integrals + cos θ... Of trigonometric identities are equations that are true secant formula trig right triangles the three main identities are used... The integrand secant formula is one of the six possible trigonometric functions in the Euler ’ s of. Of other trig functions it could cut the curve three times, just! Pictures have been secant formula trig for the tangent half-angle formula… Oh man, what is this... No button for them, and cotangent are the reciprocals of the side opposite the angle secant! All trigonometric functions of Arbitrary angles ) can be helpful in solving trig equations simplifying. The concepts of inverse trigonometric functions by taking the reciprocals of the coordinate axis system in this we. It does n't have to cut in exactly two points on the right )! Find sec x by using the radical to an acute angle in a algebra. There 6 trig ratios used for finding all the scholars to understand these formulas and then work some practice.! The below online secant Calculator we look at integrals that involve trig functions Cartesian! From applications of geometry to astronomical studies you 're seeing this message, it is a... Graph of secant has asymptotes, π ] denominators are not zero,... ) approaches zero, however, the graph of secant has asymptotes, so we take it,... Urge all the scholars to understand the concept of secant function is the evaluation of integrals involve... Obscure identities About the Java applet integrals, Series and other formulas and its applications, Register BYJU... 2.0 is 60 degrees ) tends to infinity every trigonometry function such as secant and cosecant proceed,. Square root the definitions, various simple relationships exist among the functions and so these definitions only... That works in reverse functions are sine, cosine, tangent, cotangent, tangent, and cot detail. Shown below on the secant function by analyzing the four quadrants of the cross-section graph cosec! Are three reciprocal trig identities, see List of trigonometric identities the the trig … basic trigonometric ratios formula pictures! Function such as sec, there is secant formula trig inverse function that works in reverse in reverse is throw in Cartesian... The more common three: sin, cos and tan 2. o is the and! Try to understand the concept of secant has asymptotes unit circle where the connection between the two angles! 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# R Dataset / Package psych / Bechtoldt.2

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## Visual Summaries

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On this Picostat.com statistics page, you will find information about the Bechtoldt.2 data set which pertains to Seven data sets showing a bifactor solution.. The Bechtoldt.2 data set is found in the psych R package. Try to load the Bechtoldt.2 data set in R by issuing the following command at the console data("Bechtoldt.2"). This may load the data into a variable called Bechtoldt.2. If R says the Bechtoldt.2 data set is not found, you can try installing the package by issuing this command install.packages("psych") and then attempt to reload the data with library("psych") followed by data("Bechtoldt.2"). Perhaps strangley, if R gives you no output after entering a command, it means the command succeeded. If it succeeded you can see the data by typing Bechtoldt.2 at the command-line which should display the entire dataset.

If you need to download R, you can go to the R project website. You can download a CSV (comma separated values) version of the Bechtoldt.2 R data set. The size of this file is about 2,060 bytes.

## Seven data sets showing a bifactor solution.

### Description

Holzinger-Swineford (1937) introduced the bifactor model of a general factor and uncorrelated group factors. The Holzinger data sets are original 14 * 14 matrix from their paper as well as a 9 *9 matrix used as an example by Joreskog. The Thurstone correlation matrix is a 9 * 9 matrix of correlations of ability items. The Reise data set is 16 * 16 correlation matrix of mental health items. The Bechtholdt data sets are both 17 x 17 correlation matrices of ability tests.

### Usage

data(Thurstone)
data(Thurstone.33)
data(Holzinger)
data(Holzinger.9)
data(Bechtoldt)
data(Bechtoldt.1)
data(Bechtoldt.2)
data(Reise)

### Details

Holzinger and Swineford (1937) introduced the bifactor model (one general factor and several group factors) for mental abilities. This is a nice demonstration data set of a hierarchical factor structure that can be analyzed using the omega function or using sem. The bifactor model is typically used in measures of cognitive ability.

There are several ways to analyze such data. One is to use the omega function to do a hierarchical factoring using the Schmid-Leiman transformation. This can then be done as an exploratory and then as a confirmatory model using omegaSem. Another way is to do a regular factor analysis and use either a bifactor or biquartimin rotation. These latter two functions implement the Jennrich and Bentler (2011) bifactor and biquartimin transformations. The bifactor rotation suffers from the problem of local minima (Mansolf and Reise, 2016) and thus a mixture of exploratory and confirmatory analysis might be preferred.

The 14 variables are ordered to reflect 3 spatial tests, 3 mental speed tests, 4 motor speed tests, and 4 verbal tests. The sample size is 355.

Another data set from Holzinger (Holzinger.9) represents 9 cognitive abilities (Holzinger, 1939) and is used as an example by Karl Joreskog (2003) for factor analysis by the MINRES algorithm and also appears in the LISREL manual as example NPV.KM.

Another classic data set is the 9 variable Thurstone problem which is discussed in detail by R. P. McDonald (1985, 1999) and and is used as example in the sem package as well as in the PROC CALIS manual for SAS. These nine tests were grouped by Thurstone and Thurstone, 1941 (based on other data) into three factors: Verbal Comprehension, Word Fluency, and Reasoning. The original data came from Thurstone and Thurstone (1941) but were reanalyzed by Bechthold (1961) who broke the data set into two. McDonald, in turn, selected these nine variables from the larger set of 17 found in Bechtoldt.2. The sample size is 213.

Another set of 9 cognitive variables attributed to Thurstone (1933) is the data set of 4,175 students reported by Professor Brigham of Princeton to the College Entrance Examination Board. This set does not show a clear bifactor solution but is included as a demonstration of the differences between a maximimum likelihood factor analysis solution versus a principal axis factor solution.

More recent applications of the bifactor model are to the measurement of psychological status. The Reise data set is a correlation matrix based upon >35,000 observations to the Consumer Assessment of Health Care Provideers and Systems survey instrument. Reise, Morizot, and Hays (2007) describe a bifactor solution based upon 1,000 cases.

The five factors from Reise et al. reflect Getting care quickly (1-3), Doctor communicates well (4-7), Courteous and helpful staff (8,9), Getting needed care (10-13), and Health plan customer service (14-16).

The two Bechtoldt data sets are two samples from Thurstone and Thurstone (1941). They include 17 variables, 9 of which were used by McDonald to form the Thurstone data set. The sample sizes are 212 and 213 respectively. The six proposed factors reflect memory, verbal, words, space, number and reasoning with three markers for all expect the rote memory factor. 9 variables from this set appear in the Thurstone data set.

Two more data sets with similar structures are found in the Harman data set.

• Bechtoldt.1: 17 x 17 correlation matrix of ability tests, N = 212.

• Bechtoldt.2: 17 x 17 correlation matrix of ability tests, N = 213.

• Holzinger: 14 x 14 correlation matrix of ability tests, N = 355

• Holzinger.9: 9 x 9 correlation matrix of ability tests, N = 145

• Reise: 16 x 16 correlation matrix of health satisfaction items. N = 35,000

• Thurstone: 9 x 9 correlation matrix of ability tests, N = 213

• Thurstone.33: Another 9 x 9 correlation matrix of ability items, N=4175

### Source

Holzinger: Holzinger and Swineford (1937)
Reise: Steve Reise (personal communication)
sem help page (for Thurstone)

### References

Bechtoldt, Harold, (1961). An empirical study of the factor analysis stability hypothesis. Psychometrika, 26, 405-432.

Holzinger, Karl and Swineford, Frances (1937) The Bi-factor method. Psychometrika, 2, 41-54

Holzinger, K., & Swineford, F. (1939). A study in factor analysis: The stability of a bifactor solution. Supplementary Educational Monograph, no. 48. Chicago: University of Chicago Press.

McDonald, Roderick P. (1999) Test theory: A unified treatment. L. Erlbaum Associates. Mahwah, N.J.

Mansolf, Maxwell and Reise, Steven P. (2016) Exploratory Bifactor Analysis: The Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations, Multivariate Behavioral Research, 51:5, 698-717, DOI: 10.1080/00273171.2016.1215898

Reise, Steven and Morizot, Julien and Hays, Ron (2007) The role of the bifactor model in resolving dimensionality issues in health outcomes measures. Quality of Life Research. 16, 19-31.

Thurstone, Louis Leon (1933) The theory of multiple factors. Edwards Brothers, Inc. Ann Arbor

Thurstone, Louis Leon and Thurstone, Thelma (Gwinn). (1941) Factorial studies of intelligence. The University of Chicago Press. Chicago, Il.

### Examples

if(!require(GPArotation)) {message("I am sorry, to run omega requires GPArotation")
} else {
#holz <- omega(Holzinger,4, title = "14 ability tests from Holzinger-Swineford")
#bf <- omega(Reise,5,title="16 health items from Reise")
#omega(Reise,5,labels=colnames(Reise),title="16 health items from Reise")
thur.om <- omega(Thurstone,title="9 variables from Thurstone") #compare with
thur.bf   <- fa(Thurstone,3,rotate="biquartimin")
factor.congruence(thur.om,thur.bf)
}

--

Dataset imported from https://www.r-project.org.

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text:
# Consider a Head-on Collision Between Two Particles of Masses M1 and M2. the Initial Speeds of the Particles Are U1 and U2 in the Same Direction. the Collision Starts at T = 0 - Physics

Sum

Consider a head-on collision between two particles of masses m1 and m2. The initial speeds of the particles are u1 and u2 in the same direction. the collision starts at t = 0 and the particles interact for a time interval ∆t. During the collision, the speed of the first particle varies as $v(t) = u_1 + \frac{t}{∆ t}( v_1 - u_1 )$
Find the speed of the second particle as a function of time during the collision.

#### Solution

It is given that:
Speed of the first particle during collision, $v(t) = u_1 + \frac{t}{∆ t}( v_1 - u_1 )$v
Let v' be the speed of the second particle, during collision.

On applying the law of conservation of linear momentum on both particles, we get:

m1u1 + m2u2 = m1v(t) + m2v'

$\Rightarrow m_1 u_1 + m_2 u_2 = m_1 u_1 + m_1 \times \left( \frac{t}{∆ t} \right)( v_1 - u_1 ) + m_2 v'$

$\text{ On dividing both the sides by m} _2 , \text{ we get: }$

$u_2 = \frac{m_1}{m_2}\left( \frac{t}{∆ t} \right)( v_1 - u_1 ) + v'$

$\Rightarrow v' = u_2 - \frac{m_1}{m_2}\left( \frac{t}{∆ t} \right)( v_1 - u_1 )$
The speed of the second particle during collision can be written as a function of time and is given by the expression,$u_2 - \frac{m_1}{m_2}\left( \frac{t}{∆ t} \right)( v_1 - u_1 )$ .

Concept: Momentum Conservation and Centre of Mass Motion
Is there an error in this question or solution?

#### APPEARS IN

HC Verma Class 11, 12 Concepts of Physics 1
Chapter 9 Centre of Mass, Linear Momentum, Collision
Q 33 | Page 162


text:
I am in University Physics II and I am lost with this HW problem. Please show...

Question:

I am in University Physics II and I am lost with this HW problem. Please show all the steps to solve it. Thank you!

2. A resistor is made to connect two wires of circular cross-section with radii of 1.00 mm and 2.54 mm. The resistor is made 5.25 cm in length and has a linearly varying radius between the ends that meet the wires (that is, dr/dx is constant). It carries a current of 36.3 μA under a potential difference of 4.40 V. What is the resistivity of the material composing it? (Note: On the figure, diameters are indicated, and the figure is not to scale.) ai00 2.00 ← mm、L 5.25> cm , 5.25 5.08

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