Find the Exact Value cot((27pi)/2)

Problem

cot((27*π)/2)

Solution

1. Simplify the angle by removing full rotations of 2*π to find the coterminal angle within the interval [0,2*π)

2. Divide the coefficient 27 by 2 to express the angle in terms of π

(27*π)/2=13.5*π

3. Subtract the largest even multiple of π (which is 12*π to find the coterminal angle.

13.5*π−12*π=1.5*π

4. Rewrite the simplified angle as a fraction.

1.5*π=(3*π)/2

5. Apply the definition of the cotangent function in terms of cosine and sine.

cot((3*π)/2)=cos((3*π)/2)/sin((3*π)/2)

6. Evaluate the trigonometric functions at the quadrantal angle (3*π)/2

cos((3*π)/2)=0

sin((3*π)/2)=−1

7. Calculate the final ratio.

0/(−1)=0

Final Answer

cot((27*π)/2)=0

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