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

Question

Quizplus · Accepted Answer

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

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

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