Find the exact value of the composition.
-$\sin \left( \tan ^ { - 1 } ( 2 ) \right)$
A) $2 \sqrt { 5 }$
B) $\frac { 2 \sqrt { 5 } } { 5 }$
C) $\frac { 5 \sqrt { 2 } } { 2 }$
D) $5 \sqrt { 2 }$

Question

Quizplus · Accepted Answer

To find the exact value of $\sin(\tan^{-1}(2))$, we consider a right triangle where the opposite side to the angle is 2 units and the adjacent side is 1 unit (since $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = 2$). The hypotenuse, using the Pythagorean theorem, is $\sqrt{2^2 + 1^2} = \sqrt{5}$. Therefore, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{\sqrt{5}}$, which simplifies to $\frac{2\sqrt{5}}{5}$ when rationalized.

Find the Exact Value of the Composition sin⁡(tan⁡−1(2))\sin \left( \tan ^ { - 1 } ( 2 ) \right)sin(tan−1(2))

Find the Exact Value of the Composition $\sin \left( \tan ^ { - 1 } ( 2 ) \right)$