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: $\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}$ $\cos(\frac{\theta}{2})=\pm\sqrt{\frac{1 + \cos\theta}{2}}$ $\tan(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$ $\cos195^{\circ}=-\frac{\sqrt{?+\sqrt{}}}{}$

Explanation:

Step1: Express \(195^{\circ}\) as a half - angle

Since \(195^{\circ}=\frac{390^{\circ}}{2}\), and \(\cos195^{\circ}=\cos\frac{390^{\circ}}{2}\). Also, \(195^{\circ}\) is in the third quadrant where \(\cos\) is negative.
We use the half - angle formula \(\cos\frac{\theta}{2}=-\sqrt{\frac{1 + \cos\theta}{2}}\) (negative because of the quadrant). Here \(\theta = 390^{\circ}\), and \(\cos390^{\circ}=\cos(360^{\circ}+30^{\circ})=\cos30^{\circ}=\frac{\sqrt{3}}{2}\).

Step2: Substitute into the half - angle formula

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Answer:

\(\cos195^{\circ}=-\frac{\sqrt{2+\sqrt{3}}}{2}\), so the number in the first box is \(2\), the number in the second box is \(3\), and the number in the third box is \(2\)