Find the Exact Value cos(arctan(5/2))

Problem

cos(arctan(5/2))

Solution

1. Define a variable θ such that θ=arctan(5/2)

2. Rewrite the expression using the definition of the inverse tangent function, which implies tan(θ)=5/2 for −π/2<θ<π/2

3. Identify the components of a right triangle where tan(θ)=opposite/adjacent meaning the opposite side is 5 and the adjacent side is 2

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

r=√(,5+2)

r=√(,25+4)

r=√(,29)

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

cos(θ)=2/√(,29)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,29)

cos(θ)=(2√(,29))/29

Final Answer

cos(arctan(5/2))=(2√(,29))/29

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