Question

find the exact value of the following expression.

\\( \sin \frac { 5 \pi } { 12 } \\)

the exact value of \\( \sin \frac { 5 \pi } { 12 } \\) is \\( \square \\).
(simplify your answer, including any radicals. use integers or fractions for any numbers in the ex

Explanation:

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

\(\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}\)

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

Here \(A=\frac{\pi}{4}\), \(B = \frac{\pi}{6}\)
\(\sin(\frac{\pi}{4}+\frac{\pi}{6})=\sin\frac{\pi}{4}\cos\frac{\pi}{6}+\cos\frac{\pi}{4}\sin\frac{\pi}{6}\)

Step3: Substitute the values of trigonometric functions

We know that \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\sin\frac{\pi}{6}=\frac{1}{2}\), \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\)

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Answer:

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