Question

find the exact value of the following expression.

\\( \sin \frac { 17 \pi } { 12 } \\)

\\( \sin \frac { 17 \pi } { 12 } = \\)
(simplify your answer, including any radicals. use integers or fraction

Explanation:

Step1: Transform the angle

Express \(\frac{17\pi}{12}\) as \(\pi+\frac{5\pi}{12}\). Then \(\sin\frac{17\pi}{12}=\sin(\pi + \frac{5\pi}{12})\).
According to the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\), here \(A=\pi\), \(B = \frac{5\pi}{12}\), and \(\sin(\pi+\alpha)=-\sin\alpha\) (because \(\sin\pi = 0\), \(\cos\pi=- 1\)), so \(\sin(\pi+\frac{5\pi}{12})=-\sin\frac{5\pi}{12}\).
Also, \(\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}\).

Step2: Use the sine addition formula

According to the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\), where \(A=\frac{\pi}{4}\), \(B=\frac{\pi}{6}\).
\(\sin(\frac{\pi}{4}+\frac{\pi}{6})=\sin\frac{\pi}{4}\cos\frac{\pi}{6}+\cos\frac{\pi}{4}\sin\frac{\pi}{6}\).
We know that \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\sin\frac{\pi}{6}=\frac{1}{2}\), \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\).
Substitute these values into the formula: \(\sin(\frac{\pi}{4}+\frac{\pi}{6})=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\).

Step3: Get the final result

Since \(\sin\frac{17\pi}{12}=-\sin\frac{5\pi}{12}\), then \(\sin\frac{17\pi}{12}=-\frac{\sqrt{6}+\sqrt{2}}{4}\).

Answer:

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