Find the Exact Value cos(arctan(3/4))

Problem

cos(arctan(3/4))

Solution

1. Define the angle θ such that θ=arctan(3/4)

2. Interpret the definition of the arctangent function, which implies tan(θ)=3/4 where −π/2<θ<π/2

3. Identify the sides of a right triangle using the definition tan(θ)=opposite/adjacent where the opposite side is 3 and the adjacent side is 4

4. Calculate the hypotenuse r using the Pythagorean theorem r=√(,3+4)

r=√(,9+16)

r=√(,25)

r=5

5. Evaluate the cosine of the angle using the ratio cos(θ)=adjacent/hypotenuse

cos(θ)=4/5

Final Answer

cos(arctan(3/4))=4/5

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