What is the value of $\cos 15 °$ ?

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Solution

Find the value of $\cos 15 °$ .
We know that,
$\begin{array}{rcl} \cos (15 °) & = & \cos (45 ° - 30 °) \\ = & \cos 45 ° \cos 30 ° + \sin 45 ° \sin 30 ° \{∵ \cos (A - B) = \cos A \cos B + \sin A \sin B\} \\ = & (\frac{1}{\sqrt{2}}) (\frac{\sqrt{3}}{2}) + (\frac{1}{\sqrt{2}}) (\frac{1}{2}) \{Put \cos 45 ° = \frac{1}{\sqrt{2}}, \cos 30 ° = \frac{\sqrt{3}}{2}, \sin 45 ° = \frac{1}{\sqrt{2}} \sin 30 ° = \frac{1}{2}\} \\ = & \frac{\sqrt{3}}{2 \sqrt{2}} + \frac{1}{2 \sqrt{2}} \\ = & \frac{\sqrt{3} + 1}{2 \sqrt{2}} \end{array}$
Hence, the required value is $\cos (15 °) = \frac{\sqrt{3} + 1}{2 \sqrt{2}}$

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