Question

use the reference angle to find the exact value of the given expression.
cos\frac{\pi}{4}
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Explanation:

Step1: Determine the quadrant of the angle

The angle \(\frac{\pi}{4}\) lies in the first quadrant (\(0 < \frac{\pi}{4}<\frac{\pi}{2}\)). In the first quadrant, the cosine function is positive.

Step2: Recall the reference - angle concept

For an angle \(\theta=\frac{\pi}{4}\) in the first quadrant, the reference angle \(\theta_{r}=\theta\).

Step3: Use the unit - circle value

On the unit circle, for an angle \(\theta = \frac{\pi}{4}\), \(\cos\theta=\frac{\sqrt{2}}{2}\). Since \(\cos\theta=\cos\theta_{r}\) when \(\theta\) is in the first quadrant, \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\).

Answer:

\(\frac{\sqrt{2}}{2}\)