Write the trigonometric expression as an algebraic expression in u.
-$\sin \left( \tan ^ { - 1 } u \right)$
A) $\frac { 1 } { \sqrt { u ^ { 2 } + 1 } }$
B) $\mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 }$
C) $\frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } + 1 } }$
D) $\frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } - 1 } }$

Question

Quizplus · Accepted Answer

To express $\sin(\tan^{-1}u)$ as an algebraic expression, consider a right triangle where the opposite side is $u$ and the adjacent side is $1$. The tangent of an angle is the ratio of the opposite side to the adjacent side, so $\tan(\theta) = u/1 = u$. To find $\sin(\theta)$, we need the ratio of the opposite side to the hypotenuse. The hypotenuse can be found using the Pythagorean theorem: $\sqrt{u^2 + 1^2} = \sqrt{u^2 + 1}$. Therefore, $\sin(\theta) = \frac{u}{\sqrt{u^2 + 1}}$, making option C correct.

Write the Trigonometric Expression as an Algebraic Expression in U sin⁡(tan⁡−1u)\sin \left( \tan ^ { - 1 } u \right)sin(tan−1u)

Write the Trigonometric Expression as an Algebraic Expression in U $\sin \left( \tan ^ { - 1 } u \right)$