Using sum and difference identities, what is the exact value of $\sin 165\circ$?

• A. $\frac{\sqrt{2}-\sqrt{6}}{4}$
• B. $\frac{\sqrt{3}+1}{2}$
• C. $\frac{\sqrt{6}+\sqrt{2}}{4}$
• D. $\frac{\sqrt{6}-\sqrt{2}}{4}$

Answer: (C)

Verified step by step guidance

1. Recognize that 165° can be expressed as the sum of two special angles whose sine and cosine values are well known. For example, 165° = 120° + 45° or 165° = 180° - 15°.
2. Choose the sum identity for sine: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\). Here, let \(A = 120^\circ\) and \(B = 45^\circ\).
3. Write down the exact values for sine and cosine of 120° and 45° using known special angle values: \(\sin 120^\circ = \frac{\sqrt{3}}{2}\), \(\cos 120^\circ = -\frac{1}{2}\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), and \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
4. Substitute these values into the sum identity formula: \(\sin 165^\circ = \sin 120^\circ \cos 45^\circ + \cos 120^\circ \sin 45^\circ = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(-\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\).
5. Simplify the expression by multiplying and combining like terms under a common denominator to get the exact value of \(\sin 165^\circ\) in simplest radical form.