Question

find the exact value of the expression given below.

\\( \cos \left( \frac { 5 \pi } { 12 } \
ight) \\)

\\( \cos \left( \frac { 5 \pi } { 12 } \
ight) = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Express \(\frac{5\pi}{12}\) as a sum of known angles

We know that \(\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}\).
So, \(\cos(\frac{5\pi}{12})=\cos(\frac{\pi}{4}+\frac{\pi}{6})\)

Step2: Use the cosine addition formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\)

Here \(A=\frac{\pi}{4}\) and \(B = \frac{\pi}{6}\)
\(\cos(\frac{\pi}{4}+\frac{\pi}{6})=\cos\frac{\pi}{4}\cos\frac{\pi}{6}-\sin\frac{\pi}{4}\sin\frac{\pi}{6}\)
We know that \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\sin\frac{\pi}{6}=\frac{1}{2}\)
Substitute these values:

$$
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$$

Answer:

\(\frac{\sqrt{6}-\sqrt{2}}{4}\)