Find the Exact Value cot((13pi)/3)

Problem

cot((13*π)/3)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π until the angle is within the interval [0,2*π)

(13*π)/3−4*π=(13*π)/3−(12*π)/3

(13*π)/3−(12*π)/3=π/3

2. Identify the trigonometric identity for cotangent in terms of sine and cosine.

cot(θ)=cos(θ)/sin(θ)

3. Substitute the reference angle into the identity.

cot(π/3)=cos(π/3)/sin(π/3)

4. Evaluate the trigonometric functions at the known angle π/3

cos(π/3)=1/2

sin(π/3)=√(,3)/2

5. Simplify the fraction to find the exact value.

1/2/√(,3)/2=1/√(,3)

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

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

cot((13*π)/3)=√(,3)/3

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