What is the exact value of $\sin (345°)$?

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

Answer: (C)

Verified step by step guidance

1. Recognize that the angle 345° can be expressed in terms of a related angle within the first quadrant by using the identity for sine of a difference: \(\sin(345^\circ) = \sin(360^\circ - 15^\circ)\).
2. Use the sine difference identity: \(\sin(a - b) = \sin a \cos b - \cos a \sin b\). Here, let \(a = 360^\circ\) and \(b = 15^\circ\), so \(\sin(345^\circ) = \sin(360^\circ) \cos(15^\circ) - \cos(360^\circ) \sin(15^\circ)\).
3. Recall the exact values: \(\sin(360^\circ) = 0\) and \(\cos(360^\circ) = 1\). Substitute these into the expression to simplify it to \(\sin(345^\circ) = 0 \cdot \cos(15^\circ) - 1 \cdot \sin(15^\circ) = -\sin(15^\circ)\).
4. Find the exact value of \(\sin(15^\circ)\) using the sine difference formula again: \(\sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ\).
5. Substitute the known exact values: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), and \(\sin 30^\circ = \frac{1}{2}\). Then simplify the expression to get the exact value of \(\sin(15^\circ)\), and thus \(\sin(345^\circ) = -\sin(15^\circ)\).