Find the Exact Value cot((7pi)/3)

Problem

cot((7*π)/3)

Solution

1. Find the coterminal angle by subtracting 2*π from the given angle to find an equivalent angle within the interval [0,2*π)

(7*π)/3−2*π=(7*π)/3−(6*π)/3

(7*π)/3−(6*π)/3=π/3

2. Apply the cotangent definition using the sine and cosine values of the coterminal angle.

cot(π/3)=cos(π/3)/sin(π/3)

3. Substitute the known values for cos(π/3)=1/2 and sin(π/3)=√(,3)/2

cot(π/3)=1/2/√(,3)/2

4. Simplify the fraction by canceling the denominators.

1/2/√(,3)/2=1/√(,3)

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

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

cot((7*π)/3)=√(,3)/3

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