Find the Exact Value csc((13pi)/4)

Problem

csc((13*π)/4)

Solution

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

(13*π)/4−2*π=(13*π)/4−(8*π)/4

(13*π)/4−(8*π)/4=(5*π)/4

2. Identify the quadrant for the angle (5*π)/4 Since π<(5*π)/4<(3*π)/2 the angle lies in Quadrant III.

3. Determine the reference angle θ′ for an angle in Quadrant III using the formula θ′=θ−π

θ′=(5*π)/4−π

θ′=π/4

4. Apply the reciprocal identity for cosecant, which is csc(θ)=1/sin(θ)

csc((5*π)/4)=1/sin((5*π)/4)

5. Evaluate the sine of the angle. In Quadrant III, sine is negative.

sin((5*π)/4)=−sin(π/4)

sin((5*π)/4)=−√(,2)/2

6. Calculate the cosecant by taking the reciprocal of the sine value.

csc((13*π)/4)=1/(−√(,2)/2)

csc((13*π)/4)=−2/√(,2)

7. Rationalize the denominator to find the exact value.

csc((13*π)/4)=−√(,2)

Final Answer

csc((13*π)/4)=−√(,2)

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