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

Problem

sec(arctan(5/12))

Solution

1. Define an angle θ such that θ=arctan(5/12) This implies that tan(θ)=5/12

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

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

r=√(,5+12)

r=√(,25+144)

r=√(,169)

r=13

4. Determine the value of sec(θ) using the ratio sec(θ)=hypotenuse/adjacent

sec(θ)=13/12

Final Answer

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

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