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 51

Chapter 2, Problem 51

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(cos 2π/3)

Verified step by step guidance

Recall that the function \( \sin^{-1}(x) \) (also called arcsine) gives the angle \( \theta \) in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) such that \( \sin(\theta) = x \).

First, evaluate \( \cos \frac{2\pi}{3} \). Use the unit circle or cosine properties to find the exact value of \( \cos \frac{2\pi}{3} \).

Once you have \( \cos \frac{2\pi}{3} = x \), rewrite the original expression as \( \sin^{-1}(x) \).

Next, find the angle \( \theta \) in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) such that \( \sin(\theta) = x \). This may involve using the identity \( \sin(\theta) = \sin(\pi - \theta) \) or considering the symmetry of sine and cosine functions.

Express the final answer as the exact angle \( \theta \) in radians that satisfies the above conditions, without using a calculator.

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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, with a principal range of [-π/2, π/2]. It is used to find angles from known sine values, and understanding its range is crucial to correctly interpreting results.

Evaluating Cosine at Special Angles

Cosine values at special angles like 2π/3 are well-known and can be found using the unit circle. For 2π/3, cos(2π/3) equals -1/2. Recognizing these values helps simplify expressions before applying inverse trigonometric functions.

Relationship Between Trigonometric Functions and Their Inverses

When evaluating expressions like sin⁻¹(cos θ), it is important to understand how the cosine value fits within the domain and range of the inverse sine function. This often involves converting cosine values to sine values or considering angle identities to find the correct angle.

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Introduction to Inverse Trig Functions

Related Practice

Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ (tan 2π/3)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 2 cos (2πx + 8π)

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Textbook Question

In Exercises 53–54, let f(x) = 2 sec x, g(x) = −2 tan x, and h(x) = 2x − π/2. Graph two periods of y = (f∘h)(x).

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Textbook Question

In Exercises 52–53, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sec(sin⁻¹ 1/x)

Textbook Question

In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2

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Textbook Question

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