Find the Exact Value cos(arctan(2))

Problem

cos(arctan(2))

Solution

1. Identify the inner expression as an angle θ=arctan(2) which implies tan(θ)=2

2. Represent the relationship using a right triangle where the side opposite to θ is 2 and the side adjacent to θ is 1 since tan(θ)=opposite/adjacent=2/1

3. Calculate the hypotenuse r using the Pythagorean theorem a2+b2=r2

1+2=r2

1+4=r2

r=√(,5)

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

cos(θ)=1/√(,5)

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

1/√(,5)⋅√(,5)/√(,5)=√(,5)/5

Final Answer

cos(arctan(2))=√(,5)/5

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