Questions: Use a half-angle formula to find the exact value of the expression. sin(5π/12) A. -(1/2) sqrt(2-sqrt(3)) B. -(1/2) sqrt(2+sqrt(3)) C. (1/2) sqrt(2+sqrt(3)) D. (1/2) sqrt(2-sqrt(3))

Solution Steps

To find the exact value of \(\sin \frac{5 \pi}{12}\) using a half-angle formula, we can use the identity for sine of a half-angle: \[ \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \] First, we need to express \(\frac{5 \pi}{12}\) in terms of a known angle. Notice that \(\frac{5 \pi}{12} = \frac{10 \pi}{24} = \frac{\pi}{2} - \frac{\pi}{6}\). Therefore, we can use the angle subtraction formula for cosine to find \(\cos \left(\frac{\pi}{2} - \frac{\pi}{6}\right)\), and then apply the half-angle formula.

We need to find the exact value of \( \sin \frac{5 \pi}{12} \). We can express this angle as: \[ \frac{5 \pi}{12} = \frac{\pi}{2} - \frac{\pi}{6} \]

Using the angle subtraction formula, we find: \[ \cos\left(\frac{\pi}{2} - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

We apply the half-angle formula for sine: \[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \] Substituting \( \theta = \frac{5 \pi}{6} \) (which is \( \pi - \frac{\pi}{6} \)): \[ \cos\left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2} \] Thus, we have: \[ \sin\left(\frac{5 \pi}{12}\right) = \sqrt{\frac{1 - \left(-\frac{\sqrt{3}}{2}\right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{1}{2} \sqrt{2 + \sqrt{3}} \]

Final Answer