Question

use an identity to write the expression as a single trigonometric function.

\\( \sqrt { \frac { 1 + \cos 82 ^ { \circ } } { 2 } } \\)

\\( \sqrt { \frac { 1 + \cos 82 ^ { \circ } } { 2 } } = \square ^ { \circ } \\) (type an exact answer.)

Explanation:

Step1: Recall the half - angle identity

The half - angle identity for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\).

Step2: Identify \(\alpha\)

In the given expression \(\sqrt{\frac{1+\cos82^{\circ}}{2}}\), we have \(\alpha = 82^{\circ}\).

Step3: Apply the half - angle identity

Since \(82^{\circ}\) is in the first quadrant (where cosine is positive), and using the half - angle identity \(\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}\) with \(\alpha = 82^{\circ}\), we get \(\cos\frac{82^{\circ}}{2}\).

Step4: Calculate \(\frac{\alpha}{2}\)

\(\frac{82^{\circ}}{2}=41^{\circ}\)

Answer:

\(41\)