What is the exact value of sin 165?

Answer 1

The exact value of (\sin 165^\circ) can be calculated using the sine subtraction formula. Since (165^\circ = 180^\circ - 15^\circ), we have:

[ \sin 165^\circ = \sin(180^\circ - 15^\circ) = \sin 15^\circ ]

The value of (\sin 15^\circ) can be derived from the formula (\sin(45^\circ - 30^\circ)), which gives:

[ \sin 15^\circ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4} ]

Thus, (\sin 165^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}).