Question

use a sum or difference formula to find the exact value of the trigonometric function.
sin 165°
sin 165° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Express \(165^{\circ}\) as a sum

Express \(165^{\circ}\) as \(120^{\circ}+45^{\circ}\). So, \(\sin165^{\circ}=\sin(120^{\circ} + 45^{\circ})\).

Step2: Apply the sum formula for sine

The sum formula for sine is \(\sin(A + B)=\sin A\cos B+\cos A\sin B\). Here \(A = 120^{\circ}\) and \(B=45^{\circ}\).
We know that \(\sin120^{\circ}=\frac{\sqrt{3}}{2}\), \(\cos120^{\circ}=-\frac{1}{2}\), \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), and \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\).
Substitute these values into the formula:

$$
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$$

Answer:

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