Find the Exact Value cos(arctan(5))

Problem

cos(arctan(5))

Solution

1. Identify the inner expression as an angle θ=arctan(5) which implies tan(θ)=5

2. Sketch a right triangle where the side opposite to θ is 5 and the side adjacent to θ is 1

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

1+5=h2

1+25=h2

h=√(,26)

4. Apply the definition of cosine, which is the ratio of the adjacent side to the hypotenuse.

cos(θ)=1/√(,26)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,26)

1/√(,26)⋅√(,26)/√(,26)=√(,26)/26

Final Answer

cos(arctan(5))=√(,26)/26

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