Question

Evaluate without using a calculator. $$\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$$

   Evaluate without using a calculator.
$$\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$$

Trigonometry

Charles P. McKeague,… 7th Edition

Chapter 4, Problem 63

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Solved by Expert Shamshad Waris

11/09/2020

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Step 1: We are given the expression $\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$ and we need to evaluate it. Show more…

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Evaluate without using a calculator. $$\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$$

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Key Concepts

Inverse Trigonometric Functions

These functions reverse the effect of the original trigonometric functions. For example, the inverse sine function (often written as sin?¹ or arcsin) returns the angle whose sine is a given number, effectively undoing the sine operation.

Principal Value Branch

Inverse trigonometric functions are defined to return values within a specific interval, known as the principal branch, to ensure that each input maps to a unique output. For the inverse sine function, the principal value is chosen from the interval [–?/2, ?/2].

Properties of the Sine Function

The sine function is periodic and many angles can share the same sine value. However, when applying the inverse sine, only angles within the principal branch are considered, which simplifies the evaluation to the unique angle in that range corresponding to the given sine value.

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