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

Problem

cos(arctan(4/3))

Solution

1. Define the angle θ such that θ=arctan(4/3) This implies that tan(θ)=4/3 where θ is in the interval (−π/2,π/2)

2. Relate the tangent function to the sides of a right triangle. Since tan(θ)=opposite/adjacent we can let the opposite side y=4 and the adjacent side x=3

3. Calculate the hypotenuse r using the Pythagorean theorem r=√(,x2+y2)

r=√(,3+4)

r=√(,9+16)

r=√(,25)=5

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

cos(θ)=x/r

cos(θ)=3/5

Final Answer

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

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