Find the Exact Value csc((9pi)/4)

Problem

csc((9*π)/4)

Solution

1. Identify the angle and find its coterminal equivalent within the interval [0,2*π) by subtracting 2*π

(9*π)/4−2*π=(9*π)/4−(8*π)/4

(9*π)/4−(8*π)/4=π/4

2. Apply the reciprocal identity for the cosecant function.

csc(π/4)=1/sin(π/4)

3. Substitute the known value for the sine of the reference angle.

sin(π/4)=√(,2)/2

4. Simplify the fraction by taking the reciprocal.

1/√(,2)/2=2/√(,2)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

(2√(,2))/2=√(,2)

Final Answer

csc((9*π)/4)=√(,2)

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