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

Question

Quizplus · Accepted Answer

To find the exact value of $\sin(\tan^{-1}(2))$, we consider a right triangle where the opposite side over the adjacent side (the definition of tangent) equals 2. Let the opposite side be 2 and the adjacent side be 1, making the hypotenuse $\sqrt{2^2 + 1^2} = \sqrt{5}$ by the Pythagorean theorem. The sine of an angle is the opposite side over the hypotenuse, so $\sin(\tan^{-1}(2)) = \frac{2}{\sqrt{5}}$, which simplifies to $\frac{2\sqrt{5}}{5}$ when rationalized.

Find the Exact Value of the Composition sin⁡(tan⁡−12)\sin \left( \tan ^ { - 1 } 2 \right)sin(tan−12)

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