Question

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

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

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 1, Problem 29

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Solved by Expert Julian Wong

03/08/2020

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

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

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Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, return the angle whose trigonometric function yields a given value. These functions reverse the process of standard trigonometric functions, taking into account the necessary restrictions to make them one-to-one.

Principal Value Range

The principal value range is the restricted interval over which an inverse trigonometric function is defined to yield a unique result. For arcsin, this range is typically [-?/2, ?/2], ensuring that every sine value maps to one unique angle within this interval.

Function Composition of Trigonometric Functions and Their Inverses

When composing a trigonometric function with its inverse, such as computing arcsin(sin x), the result is equal to x if and only if x is within the principal value range of the inverse function. For values of x outside this range, the output is the equivalent angle within the principal range that has the same sine value.

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