Questions: Find the exact value of the expression. [ cos ^-1left(sin frac7 pi6right) ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (cos ^-1left(sin frac7 pi6right)=) (Simplify your answer. Type an exact answer, using (pi) as needed. Use integers or fractions for any numbers in the expression.) B. There is no solution.

Solution Steps

To solve the expression \(\cos^{-1}\left(\sin \frac{7\pi}{6}\right)\), we first need to evaluate \(\sin \frac{7\pi}{6}\). The angle \(\frac{7\pi}{6}\) is in the third quadrant where the sine function is negative. Next, we find the reference angle and use the properties of the sine function to determine its value. Finally, we use the inverse cosine function to find the angle whose cosine is the calculated sine value.

The angle \(\frac{7\pi}{6}\) is located in the third quadrant, where the sine function is negative. The reference angle is \(\frac{\pi}{6}\). Therefore, we have: \[ \sin \frac{7\pi}{6} = -\sin \frac{\pi}{6} = -\frac{1}{2} \]

Next, we need to find the angle \(\theta\) such that: \[ \cos \theta = -\frac{1}{2} \] The angle that satisfies this equation in the range \([0, \pi]\) is: \[ \theta = \frac{2\pi}{3} \]

Thus, we can express the final result as: \[ \cos^{-1}\left(\sin \frac{7\pi}{6}\right) = \frac{2\pi}{3} \]

Final Answer