Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$
A) $3 / 10$
B) $2 \sqrt { 3 } / 5$
C) $( 3 - 4 \sqrt { 3 } ) / 10$
D) $( 3 - \sqrt { 2 } ) / 10$
E) $( 3 - 4 \sqrt { 3 } ) / 5$

Question

Quizplus · Accepted Answer

To find the exact value of $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$, we use the angle difference formula for sine: $\sin(A - B) = \sin A \cos B - \cos A \sin B$.Given $\sin^{-1}\frac{3}{5}$, we know that $\sin A = \frac{3}{5}$. For a right triangle with hypotenuse 5 and opposite side 3, the adjacent side (using Pythagoras) is 4. Thus, $\cos A = \frac{4}{5}$.Given $\cos^{-1}\frac{1}{2}$, we know that $\cos B = \frac{1}{2}$. This corresponds to the angle $60^\circ$ or $\frac{\pi}{3}$ radians, where $\sin B = \frac{\sqrt{3}}{2}$.Substituting these values into the formula gives:$$\sin(A - B) = \left(\frac{3}{5}\right)\left(\frac{1}{2}\right) - \left(\frac{4}{5}\right)\left(\frac{\sqrt{3}}{2}\right) = \frac{3}{10} - \frac{4\sqrt{3}}{10} = \frac{3 - 4\sqrt{3}}{10}$$Therefore, the exact value is $\frac{3 - 4\sqrt{3}}{10}$, which matches choice 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 $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$