Find the Exact Value cot((2pi)/3)

Problem

cot((2*π)/3)

Solution

1. Identify the reference angle by finding the distance to the nearest xaxis. Since (2*π)/3 is in the second quadrant, the reference angle is π−(2*π)/3=π/3

2. Determine the sign of the cotangent function in the second quadrant. In the second quadrant, x is negative and y is positive, so cot(θ)=x/y is negative.

3. Evaluate the cotangent of the reference angle using the unit circle or special triangles.

cot(π/3)=cos(π/3)/sin(π/3)

cot(π/3)=1/2/√(,3)/2

cot(π/3)=1/√(,3)

4. Rationalize the denominator of the reference value.

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

5. Apply the negative sign determined in step 2 to the result.

cot((2*π)/3)=−√(,3)/3

Final Answer

cot((2*π)/3)=−√(,3)/3

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