Find the Exact Value cos(arctan(5/4))

Problem

cos(arctan(5/4))

Solution

1. Define the angle θ such that θ=arctan(5/4) which implies tan(θ)=5/4 for an angle in the first quadrant.

2. Identify the sides of a right triangle where the side opposite to θ is 5 and the side adjacent to θ is 4

3. Calculate the hypotenuse h using the Pythagorean theorem a2+b2=c2

h=√(,4+5)

h=√(,16+25)

h=√(,41)

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

cos(θ)=4/√(,41)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,41)

cos(θ)=(4√(,41))/41

Final Answer

cos(arctan(5/4))=(4√(,41))/41

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