Questions: Find the exact value of sin^(-1)(-1/2).

Find the exact value of sin^(-1)(-1/2).
Transcript text: Find the exact value of $\sin ^{-1}\left(-\frac{1}{2}\right)$.
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Solution

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Solution Steps

To find the exact value of \(\sin^{-1}\left(-\frac{1}{2}\right)\), we need to determine the angle whose sine is \(-\frac{1}{2}\). The sine function is negative in the third and fourth quadrants, but the range of the inverse sine function (\(\sin^{-1}\)) is \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Therefore, we are looking for an angle in the fourth quadrant.

Step 1: Determine the Angle

To find the exact value of \(\sin^{-1}\left(-\frac{1}{2}\right)\), we need to identify the angle \(\theta\) such that \(\sin(\theta) = -\frac{1}{2}\). The sine function is negative in the third and fourth quadrants.

Step 2: Identify the Quadrant

The range of the inverse sine function, \(\sin^{-1}(x)\), is limited to \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Therefore, we are looking for an angle in the fourth quadrant where the sine value is \(-\frac{1}{2}\).

Step 3: Calculate the Angle

The angle that corresponds to \(\sin(\theta) = -\frac{1}{2}\) in the fourth quadrant is \(\theta = -\frac{\pi}{6}\). This is confirmed by the output, which is approximately \(-0.5236\) radians.

Final Answer

Thus, the exact value of \(\sin^{-1}\left(-\frac{1}{2}\right)\) is

\[ \boxed{-\frac{\pi}{6}} \]

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