Question

use an appropriate half - angle formula to find the exact value of the expression. cos(75°)

Explanation:

Step1: Express 75° as a half - angle

We know that \(75^{\circ}=\frac{150^{\circ}}{2}\), and the half - angle formula for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). Since \(75^{\circ}\) is in the first quadrant where cosine is positive, we use the positive formula. Here \(\alpha = 150^{\circ}\).

Step2: Find the value of \(\cos150^{\circ}\)

We know that \(\cos150^{\circ}=\cos(180^{\circ}- 30^{\circ})=-\cos30^{\circ}=-\frac{\sqrt{3}}{2}\).

Step3: Substitute into the half - angle formula

\(\cos75^{\circ}=\cos\frac{150^{\circ}}{2}=\sqrt{\frac{1+\cos150^{\circ}}{2}}=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\sqrt{\frac{2 - \sqrt{3}}{4}}=\frac{\sqrt{2-\sqrt{3}}}{2}\).

Answer:

\(\frac{\sqrt{2 - \sqrt{3}}}{2}\)