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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle /theta} corresponding to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \arctan({\frac {5}{3}})} marked look like?	Answer:
1) The third side length is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta } is the angle of the triangle with adjacent side of length 3.

Solution:

Step 1:

Draw the triangle associated to the angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan(\frac{5}{3}} . The triangle is a right triangle with side lengths 3, 5, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{34}} being the hypotenuse. The angle associated to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan(\frac{5}{3}} is the angle adjacent to the side of length 3.

Final Answer:
From the triangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\arctan\frac{5}{3}) = \frac{3}{\sqrt{34}} = \frac{3\sqrt{34}}{34}}

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