Question

use a half - angle identity to find the exact value.
cos 157.5°
cos 157.5° = □
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the quadrant and half - angle formula

Since \(157.5^{\circ}=\frac{315^{\circ}}{2}\), and \(157.5^{\circ}\) is in the second quadrant where \(\cos\theta<0\). The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1 + \cos\alpha}{2}}\) (negative because of the second quadrant). Here \(\alpha = 315^{\circ}\), and \(\cos315^{\circ}=\frac{\sqrt{2}}{2}\).

Step2: Substitute into the formula

Substitute \(\alpha = 315^{\circ}\) into \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}\).
We get \(\cos157.5^{\circ}=-\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}\).
Simplify the fraction inside the square root: \(\frac{1+\frac{\sqrt{2}}{2}}{2}=\frac{\frac{2 + \sqrt{2}}{2}}{2}=\frac{2+\sqrt{2}}{4}\).
So \(\cos157.5^{\circ}=-\sqrt{\frac{2+\sqrt{2}}{4}}\).

Step3: Simplify the square root

Since \(\sqrt{\frac{2+\sqrt{2}}{4}}=\frac{\sqrt{2+\sqrt{2}}}{2}\), then \(\cos157.5^{\circ}=-\frac{\sqrt{2+\sqrt{2}}}{2}\).

Answer:

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