Find the exact value of $\tan \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right)$ .
A) $\frac { 1 } { 5 }$
B) $\frac { 17 } { 12 }$
C) $\frac { 5 } { 13 }$
D) $\frac { 5 } { 12 }$
E) $\frac { 5 } { 17 }$

Question

Quizplus · Accepted Answer

To find $\tan(\sin^{-1}(\frac{5}{13}))$, we consider a right triangle where the opposite side is 5 and the hypotenuse is 13. Using the Pythagorean theorem, the adjacent side is $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$. Therefore, $\tan(\sin^{-1}(\frac{5}{13})) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}$.

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Find the Exact Value Of $\tan \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right)$