/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 143 \(\sin ^{-1} x+\sin ^{-1}(1-x)=\... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 143

\(\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x\)

Short Answer

Expert verified
The solution to the given equation is \(x = \frac{1}{2}\)

Step by step solution

01

Write down the given equation

The equation provided is \(\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x\)
02

Apply complement rule of inverse trigonometric function

As per the complement rule, \(\cos ^{-1} x\) can also be written as \(\frac{\pi}{2} - \sin^{-1} x\). This leaves us with the equation: \(\sin ^{-1} x+\sin ^{-1}(1-x) = \frac{\pi}{2} - \sin^{-1} x\)
03

Rearrange the equation

When rearranging the equation, move \(\sin ^{-1} x\) to the right side of the equation, resulting in:\(\sin ^{-1}(1-x) = \frac{\pi}{2} - (2 * \sin^{-1} x)\)
04

Apply the trigonometric identity of \(\sin ^{-1}\)

Using the trigonometric identity \(\sin ^{-1} y = \frac{\pi}{2} - \cos ^{-1} y\), one gets \(\sin ^{-1} x = \frac{\pi}{2} - \cos ^{-1} x\). Thus, this transforms the equation to: \(\cos ^{-1} x = \frac{\pi}{2} - \sin ^{-1} x\)
05

Match both sides of the equation

By comparing the both sides of the equation, it results in \(\cos ^{-1} x = \sin ^{-1} (1 - x)\). This equation is true if \(x = 1 - x\), so after solving for \(x\) we find that \(x = \frac{1}{2}\)

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