Find the Exact Value cos(arctan(5/12))

Problem

cos(arctan(5/12))

Solution

1. Identify the inner expression as an angle θ=arctan(5/12) which implies that tan(θ)=5/12

2. Relate the tangent function to the sides of a right triangle, where tan(θ)=opposite/adjacent Thus, the opposite side is 5 and the adjacent side is 12

3. Calculate the hypotenuse r using the Pythagorean theorem r=√(,opposite2+adjacent2)

4. Substitute the values into the theorem: r=√(,5+12)=√(,25+144)=√(,169)=13

5. Determine the value of cos(θ) using the ratio adjacent/hypotenuse

6. Evaluate the final ratio: cos(θ)=12/13

Final Answer

cos(arctan(5/12))=12/13

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