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: Identify the half - angle formula

The half - angle formula for cosine is \(\cos\frac{\theta}{2}=\pm\sqrt{\frac{1 + \cos\theta}{2}}\). We know that \(157.5^{\circ}=\frac{315^{\circ}}{2}\), so \(\theta = 315^{\circ}\). Also, \(157.5^{\circ}\) is in the second quadrant where cosine is negative, so we will use the negative sign in the half - angle formula.

Step2: Find \(\cos\theta\) where \(\theta = 315^{\circ}\)

We know that \(\cos315^{\circ}=\cos(360^{\circ}- 45^{\circ})\). Using the identity \(\cos(A - B)=\cos A\cos B+\sin A\sin B\) (or we can recall the unit circle value), \(\cos315^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}\) (since \(\cos(360^{\circ}-\alpha)=\cos\alpha\) and \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\)).

Step3: Substitute into the half - angle formula

Substitute \(\theta = 315^{\circ}\) into \(\cos\frac{\theta}{2}=-\sqrt{\frac{1+\cos\theta}{2}}\). We get \(\cos157.5^{\circ}=-\sqrt{\frac{1 + \cos315^{\circ}}{2}}\).
Substitute \(\cos315^{\circ}=\frac{\sqrt{2}}{2}\) into the formula:

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Answer:

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