Question

find the exact value of the following expression.

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

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

Explanation:

Step1: Use the sine subtraction formula

We know that \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\).
The sine subtraction formula is \(\sin(A - B)=\sin A\cos B-\cos A\sin B\).
Here \(A=\frac{\pi}{3}\), \(B = \frac{\pi}{4}\).
So \(\sin\frac{\pi}{12}=\sin(\frac{\pi}{3}-\frac{\pi}{4})=\sin\frac{\pi}{3}\cos\frac{\pi}{4}-\cos\frac{\pi}{3}\sin\frac{\pi}{4}\).

Step2: Substitute the values of trigonometric functions

We know that \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), \(\cos\frac{\pi}{3}=\frac{1}{2}\), \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\).
Substituting these values:

$$
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$$

Answer:

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