Find an algebraic expression equivalent to the given expression.
-$\sin \left( \tan ^ { - 1 } \frac { \mathrm { u } } { 2 } \right)$
A) $\frac { \mathrm { u } } { 2 }$
B) $\frac { u \sqrt { u ^ { 2 } + 4 } } { u ^ { 2 } + 4 }$
C) $\frac { \sqrt { \mathrm { u } ^ { 2 } - 4 } } { \mathrm { u } }$
D) $\frac { 2 } { \mathrm { u } }$

Question

Quizplus · Accepted Answer

To find the equivalent expression, consider a right triangle where the angle $ \theta = \tan^{-1}\left(\frac{u}{2}\right) $. This means that the opposite side is $ u $ and the adjacent side is $ 2 $, making the hypotenuse $ \sqrt{u^2 + 4} $ by the Pythagorean theorem. The sine of this angle, $ \sin(\theta) $, is the opposite over the hypotenuse, which gives $ \frac{u}{\sqrt{u^2 + 4}} $. Simplifying this expression by multiplying the numerator and the denominator by $ \sqrt{u^2 + 4} $ gives $ \frac{u\sqrt{u^2 + 4}}{u^2 + 4} $.

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