Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$
A) $\frac { 3 } { 10 }$

B) $\frac { 2 \sqrt { 3 } } { 5 }$

C) $\frac { 3 - 4 \sqrt { 3 } } { 10 }$

D) $\frac { 3 - \sqrt { 2 } } { 10 }$

E) $\frac { 3 - 4 \sqrt { 3 } } { 5 }$

Question

Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$
A) $\frac { 3 } { 10 }$

B) $\frac { 2 \sqrt { 3 } } { 5 }$

C) $\frac { 3 - 4 \sqrt { 3 } } { 10 }$

D) $\frac { 3 - \sqrt { 2 } } { 10 }$

E) $\frac { 3 - 4 \sqrt { 3 } } { 5 }$

Quizplus · Accepted Answer

Using the identity $\sin(a-b)=\sin a\cos b-\cos a\sin b$, we have:
$$\begin{aligned} \sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } ight)&=\sin\left(\sin^{-1}\frac{3}{5}ight)\cos\left(\cos^{-1}\frac{1}{2}ight)-\cos\left(\sin^{-1}\frac{3}{5}ight)\sin\left(\cos^{-1}\frac{1}{2}ight) \ &=\frac{3}{5}\cdot\frac{1}{2}-\sqrt{1-\left(\frac{3}{5}ight)^2}\cdot\frac{\sqrt{3}}{2} \ &=\frac{3}{10}-\frac{\sqrt{4^2-3^2}}{5\cdot 2} \ &=\frac{3-4\sqrt{3}}{10} \end{aligned}$$
Thus, the answer is $\boxed{	extbf{(C)}}$.

Find the Exact Value sin⁡(sin⁡−135−cos⁡−112)\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)sin(sin−153​−cos−121​)

Find the Exact Value $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$