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Chapter 5: Problem 44

Verifying a Trigonometric Identity Verify the identity. $$\sec ^{2} y-\cot ^{2}\left(\frac{\pi}{2}-y\right)=1$$

Short Answer

Expert verified
The given trigonometric identity, \( \sec^2{y} - \cot^2{(\frac{\pi}{2} - y)} = 1 \), is true.

Step by step solution

01

Start with the LHS of the Identity

Write down the left-hand side of the identity: \( \sec^2{y} - \cot^2{(\frac{\pi}{2} - y)} \)
02

Use Known Trigonometric Identities

We substitute for \(\cot{(\frac{\pi}{2} - y)}\) using known pi/2 identities: \( \cot{(\frac{\pi}{2} - y)} = \tan{y} \). Thus, the LHS of the identity can now be written as: \( \sec^2{y} - \tan^2{y} \)
03

Substitute for \( \tan^2{y} \)

We can use the Pythagorean identity \( \tan^2{y} = \sec^2{y} - 1 \). Substituting this into our equation, we get \( \sec^2{y} - (\sec^2{y} - 1) \)
04

Simplify the Equation

The equation simplifies to 1, which is the right-hand side of the identity. Therefore, the identity is verified

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are fundamental tools used in solving various problems relating to triangles and functions in trigonometry. These identities are equations that are true for all values of the variables involved.

For instance, the trigonometric identity used in verifying the equation in the exercise is related to the Pythagorean identity and the complementary angle theorem which expresses cotangent in terms of tangent. Understanding these fundamental identities helps students simplify complex expressions and solve trigonometric equations effectively. It's like having a cheat sheet that guides you through the maze of sine, cosine, and other trigonometric functions.
Pythagorean Identity
The Pythagorean identity is one of the cornerstones of trigonometry, derived from the Pythagorean theorem. It illustrates a fundamental relationship between the sine and cosine functions of an angle. The most common form of this identity is: \[ \sin^2{\theta} + \cos^2{\theta} = 1 \]

However, it can also be transformed to relate tangent and secant functions, as seen in the exercise: \[ \tan^2{\theta} = \sec^2{\theta} - 1 \]Using the Pythagorean identity allows us to manipulate expressions and verify complex identities by breaking them down into more manageable pieces.
Secant Function
The secant function, denoted as sec, is the reciprocal of the cosine function. In other words:\[ \sec{\theta} = \frac{1}{\cos{\theta}} \]

It can also be represented as the hypotenuse over the adjacent side in a right triangle. The secant function can take values greater than or equal to 1 or less than or equal to -1, as the cosine function takes values between -1 and 1. Understanding the characteristics of the secant function is essential for solving trigonometry problems and verifying identities like the one in our exercise.
Cotangent Function
The cotangent function, cot for short, is similar to the tangent function but instead of the ratio of the opposite side to the adjacent side, it uses the reciprocal ratio. It is defined as:\[ \cot{\theta} = \frac{1}{\tan{\theta}} = \frac{\cos{\theta}}{\sin{\theta}} \]

In the context of verifying trigonometric identities, understanding the cotangent function's properties and its relationship with the angle's complement - as shown in the exercise (\( \cot{\left(\frac{\pi}{2} - \theta\right)} = \tan{\theta} \)) - is crucial. The cotangent function helps in simplifying expressions and is particularly useful when dealing with angles exceeding 90 degrees but less than 180 degrees.

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Most popular questions from this chapter

A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by $$y=\frac{1}{12}(\cos 8 t-3 \sin 8 t)$$ where \(y\) is the displacement (in meters) and \(t\) is the time (in seconds). Find the times when the weight is at the point of equilibrium \((y=0)\) for \(0 \leq t \leq 1\).

Use a graphing utility to confirm the solutions found in Example 6 in two different ways. (a) Graph both sides of the equation and find the \(x\) -coordinates of the points at which the graphs intersect. Left side: \(y=\cos x+1\) Right side: \(y=\sin x\) (b) Graph the equation \(y=\cos x+1-\sin x\) and find the \(x\) -intercepts of the graph. Do both methods produce the same \(x\) -values? Which method do you prefer? Explain.

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\cos (\pi+x)$$.

The Mach number \(M\) of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. The Mach number is related to the apex angle \(\theta\) of the cone by \(\sin (\theta / 2)=1 / M\). (a) Use a half-angle formula to rewrite the equation in terms of cos \(\theta\). (b) Find the angle \(\theta\) that corresponds to a Mach number of 1. (c) Find the angle \(\theta\) that corresponds to a Mach number of 4.5. (d) The speed of sound is about 760 miles per hour. Determine the speed of an object with the Mach numbers from parts (b) and (c).

Use the Quadratic Formula to solve the equation in the interval \(0,2 \pi\). Then use a graphing utility to approximate the angle \(x\). $$4 \cos ^{2} x-4 \cos x-1=0$$
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