Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 39

Chapter 2, Problem 39

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)

Verified step by step guidance

Recognize that the expression is \( \cos(\sin^{-1}(\frac{\sqrt{2}}{2})) \). Here, \( \sin^{-1} \) is the inverse sine function, which gives an angle \( \theta \) such that \( \sin \theta = \frac{\sqrt{2}}{2} \).

Set \( \theta = \sin^{-1}(\frac{\sqrt{2}}{2}) \), so by definition, \( \sin \theta = \frac{\sqrt{2}}{2} \). Our goal is to find \( \cos \theta \).

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{\sqrt{2}}{2} \) into this identity to find \( \cos \theta \).

Calculate \( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left( \frac{\sqrt{2}}{2} \right)^2 \). Simplify the right side to find \( \cos^2 \theta \).

Take the square root of \( \cos^2 \theta \) to find \( \cos \theta \). Consider the range of \( \theta = \sin^{-1}(\frac{\sqrt{2}}{2}) \) to determine the correct sign (positive or negative) of \( \cos \theta \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, sin⁻¹(x), returns the angle whose sine is x. It is defined for x in the interval [-1, 1] and outputs angles in the range [-π/2, π/2]. Understanding this helps identify the angle corresponding to a given sine value.

Relationship Between Sine and Cosine

Sine and cosine are co-functions related by the Pythagorean identity: sin²θ + cos²θ = 1. Knowing one trigonometric value allows you to find the other by rearranging this identity, which is essential for evaluating expressions like cos(sin⁻¹ x).

Exact Values of Special Angles

Certain angles have well-known exact sine and cosine values, such as π/4 (45°), where sin(π/4) = cos(π/4) = √2/2. Recognizing these special angles helps in finding exact trigonometric values without a calculator.

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Related Practice

Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]

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In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −1/2 sec πx

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In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ 0.9)

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In Exercises 29–44, graph two periods of the given cosecant or secant function. y = csc(x − π)

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In Exercises 37–40, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = − 1/2 sin(πt/4 − π/2)

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Textbook Question

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = -4 cos 1/2 x

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