Find the exact value of the given expression. $\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)$
A) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question

Quizplus · Accepted Answer

Using the sine difference formula, $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$, with $a = \frac{\pi}{3}$ and $b = \frac{\pi}{4}$, we get $\sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{3} - 1}{2} \cdot \sqrt{2}$. Simplifying this gives $\frac{(\sqrt{3} - 1)(2\sqrt{2})}{8}$.

Find the Exact Value of the Given Expression sin⁡(π3−π4)\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)sin(3π​−4π​)

Find the Exact Value of the Given Expression $\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)$