Find an algebraic expression equivalent to the given expression.
-$\cos \left( \arctan \frac { \sqrt { 5 } } { u } \right)$
A) $\frac { \sqrt { 5 } } { u }$
B) $\frac { \sqrt { 5 u ^ { 2 } + 25 } } { 5 }$
C) $\frac { u \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$
D) $\frac { \sqrt { \mathrm { u } ^ { 2 } + 5 } } { 5 }$

Question

Quizplus · Accepted Answer

To find the algebraic expression equivalent to $\cos(\arctan(\frac{\sqrt{5}}{u}))$, we can use a right triangle where the opposite side over the adjacent side (O/A) equals $\frac{\sqrt{5}}{u}$. Let O = $\sqrt{5}$ and A = $u$. The hypotenuse (H) can be found using the Pythagorean theorem: $H = \sqrt{O^2 + A^2} = \sqrt{5 + u^2}$. The cosine of an angle is the adjacent side over the hypotenuse, so $\cos(\arctan(\frac{\sqrt{5}}{u})) = \frac{u}{\sqrt{u^2 + 5}}$, which simplifies to the expression in choice C.

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Find an Algebraic Expression Equivalent to the Given Expression $\cos \left( \arctan \frac { \sqrt { 5 } } { u } \right)$