8A F11 Q18

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

Foundations
1) What is the third length of the triangle?
2) What does the triangle look like with the angle $/theta$ corresponding to $\arctan({\frac {5}{3}})$ marked look like?	Answer:
1) The third side length is ${\sqrt {5^{2}+3^{2}}}={\sqrt {34}}$.
2) The triangle is a right triangle with side lengths 3, 5, and ${\sqrt {34}}$ being the hypotenuse. $\theta$ is the angle of the triangle with adjacent side of length 3.

Solution:

Step 1:
Draw the triangle associated to the angle $\arctan({\frac {5}{3}}$. The triangle is a right triangle with side lengths 3, 5, and ${\sqrt {34}}$ being the hypotenuse. The angle associated to $\arctan({\frac {5}{3}}$ is the angle adjacent to the side of length 3.

Final Answer:
From the triangle $\cos(\arctan {\frac {5}{3}})={\frac {3}{\sqrt {34}}}={\frac {3{\sqrt {34}}}{34}}$