Find the Exact Value csc((31pi)/6)

Problem

csc((31*π)/6)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (12*π)/6 from the argument to bring it within the range [0,2*π)

(31*π)/6−2*((12*π)/6)=(31*π)/6−(24*π)/6

(31*π)/6−(24*π)/6=(7*π)/6

2. Identify the reciprocal identity for the cosecant function.

csc((7*π)/6)=1/sin((7*π)/6)

3. Determine the sine value for the angle (7*π)/6 which is in the third quadrant where sine is negative.

sin((7*π)/6)=−sin(π/6)

sin((7*π)/6)=−1/2

4. Calculate the final value by taking the reciprocal of the sine value.

csc((7*π)/6)=1/(−1/2)

csc((7*π)/6)=−2

Final Answer

csc((31*π)/6)=−2

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