Use a sum or difference identity to find the exact value.
-$\sin \left( - \frac { 5 \pi } { 12 } \right)$
A) $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$
B) $\frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 }$
C) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
D) $\frac { - \sqrt { 6 } + \sqrt { 2 } } { 4 }$

Question

Quizplus · Accepted Answer

The exact value of $\sin \left( - \frac { 5 \pi } { 12 } \right)$ can be found by expressing $-\frac{5\pi}{12}$ as the difference of two angles whose sine and cosine values are known. Specifically, $-\frac{5\pi}{12} = -\frac{4\pi}{12} - \frac{\pi}{12} = -\frac{\pi}{3} - \frac{\pi}{4}$. Using the difference formula for sine, $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$, we get $\sin(-\frac{\pi}{3})\cos(-\frac{\pi}{4}) - \cos(-\frac{\pi}{3})\sin(-\frac{\pi}{4})$, which simplifies to $\frac{-\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{-\sqrt{2}}{2} = \frac{-\sqrt{6} - \sqrt{2}}{4}$.

Use a Sum or Difference Identity to Find the Exact sin⁡(−5π12)\sin \left( - \frac { 5 \pi } { 12 } \right)sin(−125π​)

Use a Sum or Difference Identity to Find the Exact $\sin \left( - \frac { 5 \pi } { 12 } \right)$