Write an algebraic expression that is equivalent to $\sin \left( \arctan \frac { x } { 9 } \right)$.
A) $\frac { 9 } { \sqrt { x ^ { 2 } + 81 } }$
B) $\frac { 9 } { x }$
C) $\frac { \sqrt { x ^ { 2 } + 81 } } { x }$
D) $\frac { \sqrt { x ^ { 2 } + 81 } } { 9 }$
E) $\frac { x } { \sqrt { x ^ { 2 } + 81 } }$

Question

Quizplus · Accepted Answer

To find the equivalent expression, consider a right triangle where the opposite side to the angle is $x$ and the adjacent side is $9$. The hypotenuse, using the Pythagorean theorem, is $\sqrt{x^2 + 9^2} = \sqrt{x^2 + 81}$. The definition of $\sin$ is opposite over hypotenuse, so $\sin(\arctan(x/9)) = \frac{x}{\sqrt{x^2 + 81}}$.

Write an Algebraic Expression That Is Equivalent To sin⁡(arctan⁡x9)\sin \left( \arctan \frac { x } { 9 } \right)sin(arctan9x​)

Write an Algebraic Expression That Is Equivalent To $\sin \left( \arctan \frac { x } { 9 } \right)$