use a half - angle formula to find the exact value of the following expression. cos 67.5° determine an…

use a half - angle formula to find the exact value of the following expression. cos 67.5° determine an appropriate half - angle formula for cosine and the measure of the angle. select the correct choice below and fill in the answer box to (type an integer or a decimal.) a. cos 67.5° = -√(1 - cos / 2) b. cos 67.5° = √(1 + cos / 2) c. cos 67.5° = √(1 - cos / 2) d. cos 67.5° = -√(1 + cos / 2) cos 67.5° = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.)

use a half - angle formula to find the exact value of the following expression. cos 67.5° determine an appropriate half - angle formula for cosine and the measure of the angle. select the correct choice below and fill in the answer box to (type an integer or a decimal.) a. cos 67.5° = -√(1 - cos / 2) b. cos 67.5° = √(1 + cos / 2) c. cos 67.5° = √(1 - cos / 2) d. cos 67.5° = -√(1 + cos / 2) cos 67.5° = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.)

Answer

Explanation:

Step1: Identify the half - angle formula

The half - angle formula for $\cos\frac{\alpha}{2}$ is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. Since $67.5^{\circ}=\frac{135^{\circ}}{2}$ and $67.5^{\circ}$ is in the first quadrant where cosine is positive, for $\alpha = 135^{\circ}$, the correct formula is $\cos67.5^{\circ}=\sqrt{\frac{1+\cos135^{\circ}}{2}}$. So the answer to the first part is B.

Step2: Substitute the value of $\cos135^{\circ}$

We know that $\cos135^{\circ}=-\frac{\sqrt{2}}{2}$. Substitute it into the formula: $\cos67.5^{\circ}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{2 - \sqrt{2}}{4}}=\frac{\sqrt{2-\sqrt{2}}}{2}$.

Answer:

B. $\cos67.5^{\circ}=\sqrt{\frac{1 + \cos135^{\circ}}{2}}$ $\cos67.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}$