Find the Exact Value cot((-7pi)/6)

Problem

cot(−(7*π)/6)

Solution

1. Find the coterminal angle by adding 2*π to the negative angle to locate its position on the unit circle.

θ=−(7*π)/6+2*π

θ=−(7*π)/6+(12*π)/6

θ=(5*π)/6

2. Identify the coordinates on the unit circle for the angle (5*π)/6 which is in the second quadrant.

x=cos((5*π)/6)=−√(,3)/2

y=sin((5*π)/6)=1/2

3. Apply the definition of the cotangent function, which is the ratio of the x-coordinate to the y-coordinate.

cot(θ)=x/y

cot((5*π)/6)=(−√(,3)/2)/1/2

4. Simplify the fraction by multiplying by the reciprocal of the denominator.

cot((5*π)/6)=−√(,3)/2⋅2/1

cot((5*π)/6)=−√(,3)

Final Answer

cot(−(7*π)/6)=−√(,3)

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