Question

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use an appropriate half - angle formula to find the exact value of the expression.
cos(105°)
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Explanation:

Step1: Express 105° as a half - angle

Since \(105^{\circ}=\frac{210^{\circ}}{2}\), we can use the half - angle formula for cosine \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). Here \(\alpha = 210^{\circ}\).

Step2: Determine the sign

The angle \(105^{\circ}\) is in the second quadrant, and in the second quadrant, \(\cos\theta<0\). So \(\cos105^{\circ}=-\sqrt{\frac{1+\cos210^{\circ}}{2}}\).

Step3: Find the value of \(\cos210^{\circ}\)

We know that \(\cos210^{\circ}=\cos(180^{\circ} + 30^{\circ})=-\cos30^{\circ}=-\frac{\sqrt{3}}{2}\).

Step4: Substitute \(\cos210^{\circ}\) into the formula

\[

$$\begin{align*} \cos105^{\circ}&=-\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}\\ &=-\sqrt{\frac{2 - \sqrt{3}}{4}}\\ &=-\frac{\sqrt{2-\sqrt{3}}}{2} \end{align*}$$

\]

Answer:

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