Find the exact value. $\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$
A) $5 / 6$
B) $12 / 5$
C) $119 / 120$
D) $120 / 169$
E) none of these

Question

Quizplus · Accepted Answer

Let $\theta = \tan^{-1} \frac{5}{12}$. Then $\tan \theta = \frac{5}{12}$. We can draw a right triangle with opposite side $5$ and adjacent side $12$ and use the Pythagorean theorem to find that the hypotenuse has length $13$. So $\sin \theta = \frac{5}{13}$ and $\cos \theta = \frac{12}{13}$. Using the double angle identity for sine, we have
$$\sin (2 \theta) = 2 \sin \theta \cos \theta = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{120}{169}.$$
Thus, our answer is $\sin (2 \tan^{-1} \frac{5}{12}) = \sin (2 \theta) = \boxed{\textbf{(D)}\ \frac{120}{169}}$.

Find the Exact Value sin⁡(2tan⁡−1512)\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)sin(2tan−1125​)

Find the Exact Value $\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$