Use reference angles to find the exact value of the following expression. Do not use a calculator. \( \csc \frac{5 \pi}{6} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \csc \frac{5 \pi}{6}=\square \) (Simplify your answer. Type an exact answer, usilig radicals as needed. Use integers or fractions for any numbers in the the denominator.)

To find the exact value of \( \csc \frac{5 \pi}{6} \), we first need to determine the sine of the angle \( \frac{5 \pi}{6} \). 1. **Identify the reference angle**: The angle \( \frac{5 \pi}{6} \) is in the second quadrant. The reference angle can be found by subtracting \( \frac{5 \pi}{6} \) from \( \pi \): \[ \pi - \frac{5 \pi}{6} = \frac{6 \pi}{6} - \frac{5 \pi}{6} = \frac{\pi}{6} \] 2. **Find the sine of the reference angle**: The sine of \( \frac{\pi}{6} \) is known: \[ \sin \frac{\pi}{6} = \frac{1}{2} \] 3. **Determine the sine in the second quadrant**: In the second quadrant, the sine function is positive. Therefore: \[ \sin \frac{5 \pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2} \] 4. **Find the cosecant**: The cosecant is the reciprocal of the sine: \[ \csc \frac{5 \pi}{6} = \frac{1}{\sin \frac{5 \pi}{6}} = \frac{1}{\frac{1}{2}} = 2 \] Thus, the exact value of \( \csc \frac{5 \pi}{6} \) is: \[ \csc \frac{5 \pi}{6} = 2 \] So the final answer is: A. \( \csc \frac{5 \pi}{6} = 2 \)