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 43

Chapter 2, Problem 43

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)

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Recognize that the expression is \( \sin^{-1}(\sin(\frac{5\pi}{6})) \), which means we want to find the angle whose sine is \( \sin(\frac{5\pi}{6}) \).

Recall that \( \sin(\frac{5\pi}{6}) = \sin(\pi - \frac{\pi}{6}) = \sin(\frac{\pi}{6}) \), so \( \sin(\frac{5\pi}{6}) = \frac{1}{2} \).

Understand that the function \( \sin^{-1}(x) \) (also called arcsin) returns values in the principal range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Since \( \sin^{-1}(\sin(\frac{5\pi}{6})) = \sin^{-1}(\frac{1}{2}) \), find the angle in the principal range whose sine is \( \frac{1}{2} \).

Identify that the angle in \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) with sine \( \frac{1}{2} \) is \( \frac{\pi}{6} \), so \( \sin^{-1}(\sin(\frac{5\pi}{6})) = \frac{\pi}{6} \).

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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, denoted sin⁻¹ or arcsin, returns the angle whose sine is a given value. Its output is restricted to the principal range of [-π/2, π/2], meaning the result must lie within this interval to be considered the principal value.

Sine Function Periodicity and Symmetry

The sine function is periodic with period 2π and is symmetric about the origin (odd function). This means sin(θ) = sin(π - θ), which helps in finding equivalent angles within the principal range when evaluating inverse sine expressions.

Evaluating sin⁻¹(sin θ) for Angles Outside the Principal Range

When θ is outside the principal range of arcsin, sin⁻¹(sin θ) equals the angle within [-π/2, π/2] that has the same sine value as θ. This often involves using angle identities or symmetry to find the equivalent angle inside the principal range.

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Related Practice

Textbook Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec(x + π)

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Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹ 125)

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Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)

Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = cos(x + π/2)

Textbook Question

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

Textbook Question

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