Find the exact value of each expression.
sin (-13π/12)

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Recognize that the angle is negative: \(-\frac{13\pi}{12}\). Use the identity \(\sin(-\theta) = -\sin(\theta)\) to rewrite the expression as \(-\sin\left(\frac{13\pi}{12}\right)\).

Express \(\frac{13\pi}{12}\) as a sum of angles whose sine and cosine values are known. For example, \(\frac{13\pi}{12} = \pi + \frac{\pi}{12}\) or \(\frac{13\pi}{12} = \frac{3\pi}{4} + \frac{\pi}{3}\).

Use the sine addition formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) to expand \(\sin\left(\frac{13\pi}{12}\right)\) based on the chosen angle decomposition.

Substitute the exact values of \(\sin\) and \(\cos\) for the known angles (like \(\frac{3\pi}{4}\) and \(\frac{\pi}{3}\)) into the expanded expression.

Simplify the expression algebraically to find the exact value of \(\sin\left(\frac{13\pi}{12}\right)\), then multiply by \(-1\) to get the value of \(\sin\left(-\frac{13\pi}{12}\right)\).

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