Use the appropriate addition formula to find the exact value of the expression.
-$\sin \left( \frac { 19 \pi } { 12 } \right)$
A) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
B) $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$
C)$ - \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$
D) $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$

Question

Quizplus · Accepted Answer

To find the exact value of $\sin \left( \frac { 19 \pi } { 12 } \right)$, we can express it as a sum of angles for which we know the exact values of sine. The expression can be rewritten as $\sin \left( \frac { 16 \pi } { 12 } + \frac { 3 \pi } { 12 } \right) = \sin \left( \frac { 4 \pi } { 3 } + \frac { \pi } { 4 } \right)$. Using the addition formula for sine, $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$, we get $\sin \left( \frac { 4 \pi } { 3 } \right) \cos \left( \frac { \pi } { 4 } \right) + \cos \left( \frac { 4 \pi } { 3 } \right) \sin \left( \frac { \pi } { 4 } \right)$. Knowing the exact values, $\sin \left( \frac { 4 \pi } { 3 } \right) = -\frac{\sqrt{3}}{2}$, $\cos \left( \frac { 4 \pi } { 3 } \right) = -\frac{1}{2}$, $\sin \left( \frac { \pi } { 4 } \right) = \frac{\sqrt{2}}{2}$, and $\cos \left( \frac { \pi } { 4 } \right) = \frac{\sqrt{2}}{2}$, we find the value to be $-\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{6} + \sqrt{2}}{4}$.

Use the Appropriate Addition Formula to Find the Exact Value sin⁡(19π12)\sin \left( \frac { 19 \pi } { 12 } \right)sin(1219π​)

Use the Appropriate Addition Formula to Find the Exact Value $\sin \left( \frac { 19 \pi } { 12 } \right)$