Solve the equation for exact solutions.
-$$\sin ^ { - 1 } x + \cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right) = \pi$$
A) $\left\{ \frac { \sqrt { 3 } } { 2 } \right\}$
B) $\{ 1 \}$
C) $\{ 0 \}$
D) $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$

Question

Quizplus · Accepted Answer

The equation can be simplified using the identity that $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ for all $x$ in the domain of these functions. Given $\cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3}$ (since the cosine of $120^\circ$ or $\frac{2\pi}{3}$ radians is $-\frac{1}{2}$), we can set up the equation $\sin^{-1}x = \pi - \frac{2\pi}{3} = \frac{\pi}{3}$. The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$, so $x = \frac{\sqrt{3}}{2}$.

Solve the Equation for Exact Solutions sin⁡−1x+cos⁡−1(−12)=π\sin ^ { - 1 } x + \cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right) = \pisin−1x+cos−1(−21​)=π

Solve the Equation for Exact Solutions $\sin ^ { - 1 } x + \cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right) = \pi$