Question

use reference angles to find the exact value of the following expression. do not use a calculator.

\\( \csc \frac { 3 \pi } { 4 } \\)

determine the reference angle for \\( \frac { 3 \pi } { 4 } \\).

the reference angle is
(type your answer in radians. use integers or fractions for any numbers in the expression. type an exact answer, using \\( \pi \\) as needed.)

Explanation:

Step1: Identify the quadrant of \( \frac{3\pi}{4} \)

The angle \( \frac{3\pi}{4} \) is in the second quadrant (since \( \frac{\pi}{2} < \frac{3\pi}{4} < \pi \)).

Step2: Calculate the reference angle

For an angle \( \theta \) in the second quadrant, the reference angle \( \theta' \) is given by \( \pi - \theta \).
So, for \( \theta = \frac{3\pi}{4} \), the reference angle \( \theta' = \pi - \frac{3\pi}{4} \).
Simplify \( \pi - \frac{3\pi}{4} = \frac{4\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{4} \).

Answer:

\( \frac{\pi}{4} \)