# 008A Sample Final A, Question 18

Question: Compute ${\displaystyle \cos(\arctan {\frac {5}{3}})}$

Foundations:
1) Arctan can be thought of as referencing an angle in a triangle. What are the side lengths of the triangle?
1) Since tangent is opposite/adjacent, the side lengths of the triangle are ${\displaystyle 3,5{\text{, and }}{\sqrt {34}}}$
${\displaystyle \arctan \left({\frac {5}{3}}\right)}$   is the measure of an angle in the triangle with side lengths ${\displaystyle 3,5{\text{, and }}{\sqrt {34}}}$. The angle that corresponds to ${\displaystyle \arctan \left({\frac {5}{3}}\right)}$ is the one between the side of length 3 and the side of length   ${\displaystyle {\sqrt {34}}}$
Now we just have to take ${\displaystyle \cos }$  of the angle referred to in step 1.
${\displaystyle \cos \left(\arctan {\frac {5}{3}}\right)={\frac {5}{\sqrt {34}}}}$