Find the Exact Value cot((4pi)/3)

Problem

cot((4*π)/3)

Solution

1. Identify the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle is in Quadrant III.

2. Determine the reference angle. For an angle θ in Quadrant III, the reference angle θ′ is calculated as θ−π

θ′=(4*π)/3−π=π/3

3. Apply the sign of the cotangent function in Quadrant III. In Quadrant III, both sine and cosine are negative, so their ratio cot(θ)=cos(θ)/sin(θ) is positive.

4. Evaluate the cotangent of the reference angle. Using the known values for special angles:

cot(π/3)=cos(π/3)/sin(π/3)

cot(π/3)=1/2/√(,3)/2

cot(π/3)=1/√(,3)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

cot((4*π)/3)=√(,3)/3

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