008A Sample Final A, Question 18

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

Foundations
1) Arctan can be thought of as referencing an angle in a triangle. What are the side lengths of the triangle?
Answer:
1) Since tangent is opposite/adjacent, the side lengths of the triangle are $3,5{\text{, and }}{\sqrt {34}}$

Solution:

Step 1:
$\arctan \left({\frac {5}{3}}\right)$   is the measure of an angle in the triangle with side lengths $3,5{\text{, and }}{\sqrt {34}}$. The angle that corresponds to $\arctan \left({\frac {5}{3}}\right)$ is the one between the side of length 3 and the side of length   ${\sqrt {34}}$

Step 2:
Now we just have to take $\cos$  of the angle referred to in step 1.

Final Answer:
$\cos \left(\arctan {\frac {5}{3}}\right)={\frac {5}{\sqrt {34}}}$

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