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

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

Solution

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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