Question

use the half - angle formulas to find the exact value of the expression cos 112.5°. which half - angle formula should be used to find the exact value of the cos 112.5°? select the correct choice below and fill in the answer boxes to complete your choice (type integers or decimals ). a. cos 112.5° = cos (\frac{^{circ}}{2}=sqrt{\frac{1 + cos^{circ}}{2}}) b. cos 112.5° = cos (\frac{225^{circ}}{2}=-sqrt{\frac{1+cos 225^{circ}}{2}}) c. cos 112.5° = cos (\frac{^{circ}}{2}=sqrt{\frac{1 - cos^{circ}}{2}}) d. cos 112.5° = cos (\frac{^{circ}}{2}=-sqrt{\frac{1 - cos^{circ}}{2}}) cos 112.5° = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression ).

Explanation:

Step1: Identify the half - angle formula

Since \(112.5^{\circ}=\frac{225^{\circ}}{2}\), and \(112.5^{\circ}\) is in the second - quadrant where cosine is negative. The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1 + \cos\alpha}{2}}\) when \(\frac{\alpha}{2}\) is in the second - quadrant. Here \(\alpha = 225^{\circ}\), so \(\cos112.5^{\circ}=\cos\frac{225^{\circ}}{2}=-\sqrt{\frac{1+\cos225^{\circ}}{2}}\).

Step2: Find the value of \(\cos225^{\circ}\)

We know that \(\cos225^{\circ}=\cos(180^{\circ}+45^{\circ})=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}\).

Step3: Substitute the value of \(\cos225^{\circ}\) into the formula

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Answer:

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