Find the Exact Value cos(arctan(-2))

Problem

cos(arctan(−2))

Solution

1. Identify the inner function as an angle θ=arctan(−2) By the definition of the inverse tangent function, this means tan(θ)=−2 where θ must be in the interval (−π/2,π/2)

2. Determine the quadrant of θ Since the tangent value is negative, θ must lie in Quadrant IV. In this quadrant, the cosine function is positive.

3. Represent the tangent relationship using a right triangle or coordinates. Let tan(θ)=y/x=(−2)/1 Thus, we can set y=−2 and x=1

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

r=√(,1+(−2)2)

r=√(,1+4)

r=√(,5)

5. Evaluate the cosine of the angle using the ratio cos(θ)=x/r

cos(θ)=1/√(,5)

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

cos(θ)=√(,5)/5

Final Answer

cos(arctan(−2))=√(,5)/5

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