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

Problem

cot((2*π)/3)

Solution

1. Identify the reference angle for (2*π)/3 in the second quadrant.

θ′=π−(2*π)/3=π/3

2. Determine the sign of the cotangent function in the second quadrant.

In Quadrant II, *cot(θ)<0

3. Apply the reference angle identity for cotangent.

cot((2*π)/3)=−cot(π/3)

4. Evaluate the cotangent of the reference angle using the ratio of cosine to sine.

cot(π/3)=cos(π/3)/sin(π/3)=1/2/√(,3)/2=1/√(,3)

5. Rationalize the denominator of the result.

1/√(,3)=√(,3)/3

6. Combine the value with the negative sign from the quadrant determination.

cot((2*π)/3)=−√(,3)/3

Final Answer

cot((2*π)/3)=−√(,3)/3

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