Find the Exact Value cos(arctan(-2/3))

Problem

cos(arctan(−2/3))

Solution

1. Identify the inner function as an angle θ=arctan(−2/3) By the definition of the inverse tangent function, this means tan(θ)=−2/3 where θ is in the interval (−π/2,π/2)

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

3. Relate the tangent to the sides of a right triangle. Let the opposite side be y=−2 and the adjacent side be x=3

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

r=√(,3+(−2)2)

r=√(,9+4)

r=√(,13)

5. Apply the definition of cosine, which is cos(θ)=x/r

cos(θ)=3/√(,13)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,13)

cos(θ)=(3√(,13))/13

Final Answer

cos(arctan(−2/3))=(3√(,13))/13

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