Solve the equation for exact solutions.
-$$\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 5 } { 13 }$$
A) $\{ 0 \}$
B) $\left\{ \frac { 12 } { 5 } \right\}$
C) $\left\{ \frac { 12 } { 13 } \right\}$
D) $\varnothing$

Question

Quizplus · Accepted Answer

Given $\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 5 } { 13 }$, we can use the identity $\sin ^ { - 1 } y = \cos ^ { - 1 } \sqrt{1 - y^2}$ to find $x$. Substituting $y = \frac{5}{13}$, we get $\cos ^ { - 1 } x = \cos ^ { - 1 } \sqrt{1 - \left(\frac{5}{13}\right)^2}$. Simplifying inside the square root gives $\sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}$. Therefore, $x = \frac{12}{13}$.

Solve the Equation for Exact Solutions cos⁡−1x=sin⁡−1513\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 5 } { 13 }cos−1x=sin−1135​

Solve the Equation for Exact Solutions $\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 5 } { 13 }$