Using sum and difference identities, what is the exact value of $\sin (105°)$?

• A. $\frac{\sqrt{2}}{2}$
• 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 105° can be expressed as the sum of two special angles whose sine and cosine values are well known. For example, 105° = 60° + 45°.
2. Recall the sine sum identity: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\). Here, let \(A = 60^\circ\) and \(B = 45^\circ\).
3. Substitute the known exact values: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), and \(\cos 45^\circ = \frac{\sqrt{2}}{2}\) into the identity.
4. Calculate each term separately: \(\sin 60^\circ \cos 45^\circ\) and \(\cos 60^\circ \sin 45^\circ\), then add them together to get \(\sin 105^\circ\).
5. Simplify the resulting expression by combining like terms and rationalizing if necessary to express the exact value in simplest radical form.