Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sec⁻¹ (−1)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
4:17 minutesProblem 43
Textbook Question
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)
Verified step by step guidance1
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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Video duration:
4mKey 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.
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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.
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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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