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Chapter 5: Problem 107

If \(\sin ^{-1}\left(\sin \frac{\pi}{3}\right)=\frac{\pi}{3},\) is \(\sin ^{-1}\left(\sin \frac{5 \pi}{6}\right)=\frac{5 \pi}{6} ?\) Explain your answer.

Short Answer

Expert verified
\(\sin^{-1}\left(\sin(5\pi/6)\right) = \pi/6\), not \(5\pi/6\).

Step by step solution

01

Understand the Inverse Sine Function

The inverse sine function, denoted as \(\sin^{-1}(x)\) or \(arcsin(x)\), undoes the sine function. It is defined for values between \(-1\) and \(1\) and has a range of \(-\pi/2\) to \(\pi/2\).
02

Analyze the First Part of the Equation

The first part of the equation, \(\sin^{-1}\left(\sin(\pi / 3)\right)=\pi / 3\), is true due to the property of the inverse sine function. The sine of \(\pi/3\) is within the interval of \([-1,1]\) and \(\pi/3\) is within the range of \(-\pi/2\) to \(\pi/2\). So you can say \(\sin^{-1}\) of \(\sin(\pi / 3)\) is \(\pi / 3\).
03

Analyze the Second Part of the Equation

The second part of the equation is \(\sin^{-1}\left(\sin(5 \pi / 6)\right)\). The sine of \(5\pi/6\) is within the interval of \([-1,1]\), however \(5\pi/6\) is not within the range of \[-\pi/2, \pi/2\]. So you can't directly say \(\sin^{-1}\) of \(\sin(5 \pi/6)\) is \(5 \pi/6\).
04

Find the Equivalent Value in the Range

To find a valid output of the function \(\sin^{-1}\) that gives the same sine value as \(5\pi/6\), you need to check for a value in the range from \(-\pi/2\) to \(\pi/2\) that has the same sine value as \(5\pi/6\). Since \(sin(\pi -x ) = sin(x)\), the equivalent value is \(\pi - 5\pi / 6 = \pi/6\). So, the \(\sin^{-1}\left(\sin(5\pi/6)\right)\) equals to \(\pi/6\), not \(5\pi/6\).

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