Find the Exact Value cot((5pi)/3)

Problem

cot((5*π)/3)

Solution

1. Identify the reference angle by finding the distance to the nearest multiple of π Since (5*π)/3 is in the fourth quadrant, the reference angle is 2*π−(5*π)/3=π/3

2. Determine the sign of the cotangent function in the fourth quadrant. In the fourth quadrant, cosine is positive and sine is negative, so cot(θ)=cos(θ)/sin(θ) must be negative.

3. Evaluate the cotangent of the reference angle using known trigonometric values.

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. Combine the reference value with the correct sign from step 2.

cot((5*π)/3)=−√(,3)/3

Final Answer

cot((5*π)/3)=−√(,3)/3

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