Question

find the exact value without a calculator. double - angle formulas: $\sin(2\theta)=2\sin\theta\cos\theta$ $\cos(2\theta)=\cos^{2}\theta - \sin^{2}\theta$ $\tan(2\theta)=\frac{2\tan\theta}{1 - \tan^{2}\theta}$ half - angle formulas: $\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}$ $\cos(\frac{\theta}{2})=\pm\sqrt{\frac{1 + \cos\theta}{2}}$ $\tan(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$ $\cos\frac{\pi}{8}=\frac{\sqrt{?+\sqrt{}}}{}$

Explanation:

Step1: Use half - angle formula

Since \(\frac{\pi}{8}=\frac{\frac{\pi}{4}}{2}\), and for \(\cos(\frac{\theta}{2})\) with \(\theta = \frac{\pi}{4}\), the half - angle formula \(\cos(\frac{\theta}{2})=\sqrt{\frac{1 + \cos\theta}{2}}\) (because \(\frac{\pi}{8}\) is in the first quadrant where cosine is positive).

Step2: Substitute \(\theta=\frac{\pi}{4}\)

We know that \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\). Substitute into the formula: \(\cos(\frac{\pi}{8})=\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}=\frac{\sqrt{2 + \sqrt{2}}}{2}\)

Answer:

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