Question

use reference angles to find the exact value of the following expression.
cos \frac{7 \pi}{4}
determine the reference angle for \frac{7 \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: Determine the quadrant of \( \frac{7\pi}{4} \)

The angle \( \frac{7\pi}{4} \) is in the fourth quadrant because \( \frac{3\pi}{2} < \frac{7\pi}{4} < 2\pi \).

Step2: Find the reference angle

For an angle \( \theta \) in the fourth quadrant, the reference angle \( \theta' \) is given by \( 2\pi - \theta \).
So, for \( \theta = \frac{7\pi}{4} \), the reference angle \( \theta' = 2\pi - \frac{7\pi}{4} \).
Calculate \( 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4} \).

Answer:

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