Questions: Evaluate the following expression. Write the exact answer in radians. Do not round. If the answer does not exist, write DNE, [ sin ^-1(-frac12) ]

$Evaluate the following expression. Write the exact answer in radians. Do not round. If the answer does not exist, write DNE, [ sin ^-1(-frac12) ]$

Transcript text: Evaluate the following expression. Write the exact answer in radians. Do not round. If the answer does not exist, write DNE, \[ \sin ^{-1}\left(-\frac{1}{2}\right) \]

Solution

To solve the given expression $\sin^{-1}\left(-\frac{1}{2}\right)$, we need to find the angle whose sine is $-\frac{1}{2}$. The range of the inverse sine function, $\sin^{-1}(x)$, is $[- \frac{\pi}{2}, \frac{\pi}{2}]$. We know that $\sin(-\frac{\pi}{6}) = -\frac{1}{2}$.

Solution Approach

Identify the angle in the range $[- \frac{\pi}{2}, \frac{\pi}{2}]$ whose sine is $-\frac{1}{2}$.
Use the inverse sine function to find this angle.

Step 1: Identify the Inverse Sine

We need to evaluate the expression $ \sin^{-1}\left(-\frac{1}{2}\right) $. This expression represents the angle $ \theta $ such that $ \sin(\theta) = -\frac{1}{2} $.

Step 2: Determine the Angle

The sine function is negative in the fourth quadrant. The reference angle for which $ \sin(\theta) = \frac{1}{2} $ is $ \frac{\pi}{6} $. Therefore, the angle in the fourth quadrant that corresponds to $ \sin(\theta) = -\frac{1}{2} $ is given by: \[ \theta = -\frac{\pi}{6} \]

Step 3: Final Result

Thus, the exact answer in radians for $ \sin^{-1}\left(-\frac{1}{2}\right) $ is: \[ \boxed{-\frac{\pi}{6}} \]

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