Find the exact value of each expression.
sin (- 5π/12)

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Recognize that the angle is negative: \(-\frac{5\pi}{12}\). Use the identity \(\sin(-\theta) = -\sin(\theta)\) to rewrite the expression as \(-\sin\left(\frac{5\pi}{12}\right)\).

Express \(\frac{5\pi}{12}\) as a sum of angles whose sine and cosine values are known. For example, \(\frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}\).

Use the sine addition formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). Substitute \(a = \frac{\pi}{3}\) and \(b = \frac{\pi}{4}\) to get \(\sin\left(\frac{5\pi}{12}\right) = \sin\frac{\pi}{3} \cos\frac{\pi}{4} + \cos\frac{\pi}{3} \sin\frac{\pi}{4}\).

Recall the exact values: \(\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\), \(\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{3} = \frac{1}{2}\), and \(\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\). Substitute these into the expression.

Combine the terms and simplify the expression inside the parentheses, then apply the negative sign from step 1 to find the exact value of \(\sin\left(-\frac{5\pi}{12}\right)\).

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