Question

use a half - angle identity to find the exact value.
cos 22.5°
cos 22.5° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressi

Explanation:

Step1: Apply the half - angle identity

The half - angle identity for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$.
Since \(22.5^{\circ}=\frac{45^{\circ}}{2}\) and \(22.5^{\circ}\) is in the first quadrant (where cosine is positive), we use the positive form. So, \(\cos22.5^{\circ}=\sqrt{\frac{1+\cos45^{\circ}}{2}}\).

Step2: Substitute the value of \(\cos45^{\circ}\)

We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\). Substituting this into the formula:

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Answer:

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