Find the Exact Value csc((8pi)/3)

Problem

csc((8*π)/3)

Solution

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

(8*π)/3−2*π=(8*π)/3−(6*π)/3

(8*π)/3−(6*π)/3=(2*π)/3

2. Identify the reciprocal identity for the cosecant function.

csc((2*π)/3)=1/sin((2*π)/3)

3. Determine the sine value for the angle (2*π)/3 which is in the second quadrant where sine is positive.

sin((2*π)/3)=√(,3)/2

4. Calculate the reciprocal to find the value of the cosecant function.

csc((2*π)/3)=1/√(,3)/2

csc((2*π)/3)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

csc((8*π)/3)=(2√(,3))/3

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