Question

use one or more of the six sum and difference identities to find the exact value of the expression.
\\( \sin \left(15^{\circ}\
ight) \\)
find the exact value of the expression.
\\( \sin \left(15^{\circ}\
ight)= \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expres

Explanation:

Step1: Express \(15^{\circ}\) as a difference of two known angles

We know that \(15^{\circ}=45^{\circ} - 30^{\circ}\). So, \(\sin(15^{\circ})=\sin(45^{\circ}-30^{\circ})\).

Step2: Apply the sine - difference identity

The sine - difference identity is \(\sin(A - B)=\sin A\cos B-\cos A\sin B\). Here \(A = 45^{\circ}\) and \(B=30^{\circ}\).
We know that \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), \(\sin30^{\circ}=\frac{1}{2}\), and \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).
Substitute these values into the formula:

$$
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$$

Answer:

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