Find the exact value of the expression, if possible. Do not use a calculator.
-$\sin \left( \tan ^ { - 1 } \frac { x } { \sqrt { 2 } } \right)$
A) $\frac { x \sqrt { x ^ { 2 } + 2 } } { x ^ { 2 } + 2 }$
B) $\frac { x \sqrt { x ^ { 2 } - 2 } } { x ^ { 2 } - 2 }$
C) $x \sqrt { x ^ { 2 } + 2 }$
D) $\frac { \sqrt { x ^ { 2 } + 2 } } { x ^ { 2 } + 2 }$

Question

Quizplus · Accepted Answer

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

Find the Exact Value of the Expression, If Possible sin⁡(tan⁡−1x2)\sin \left( \tan ^ { - 1 } \frac { x } { \sqrt { 2 } } \right)sin(tan−12​x​)

Find the Exact Value of the Expression, If Possible $\sin \left( \tan ^ { - 1 } \frac { x } { \sqrt { 2 } } \right)$