Find the Exact Value csc((11pi)/4)

Problem

csc((11*π)/4)

Solution

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

(11*π)/4−2*π=(11*π)/4−(8*π)/4

(11*π)/4−(8*π)/4=(3*π)/4

2. Identify the quadrant for the angle (3*π)/4 Since π/2<(3*π)/4<π the angle is in Quadrant II.

3. Determine the reference angle for (3*π)/4 in Quadrant II.

π−(3*π)/4=π/4

4. Apply the cosecant identity csc(θ)=1/sin(θ) In Quadrant II, the sine function (and thus the cosecant function) is positive.

csc((3*π)/4)=1/sin((3*π)/4)

5. Evaluate the sine of the reference angle.

sin(π/4)=√(,2)/2

6. Calculate the reciprocal to find the cosecant value.

csc((3*π)/4)=1/√(,2)/2

2/√(,2)=√(,2)

Final Answer

csc((11*π)/4)=√(,2)

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