Find the Exact Value csc((13pi)/8)

Problem

csc((13*π)/8)

Solution

1. Identify the reference angle and the quadrant. The angle (13*π)/8 is in the fourth quadrant because (3*π)/2<(13*π)/8<2*π

2. Determine the sign of the cosecant function. In the fourth quadrant, the sine function is negative, so csc((13*π)/8) is also negative.

3. Use the identity csc(θ)=1/sin(θ) We need to find sin((13*π)/8)

4. Apply the half-angle formula for sine, sin(α/2)=±√(,(1−cos(α))/2) Let α/2=(13*π)/8 so α=(13*π)/4

5. Evaluate cos((13*π)/4) Since (13*π)/4=3*π+π/4 the angle is coterminal with (5*π)/4 in the third quadrant.

6. Substitute cos((5*π)/4)=−√(,2)/2 into the half-angle formula.

7. Calculate the sine value:

sin((13*π)/8)=−√(,(1−(−√(,2)/2))/2)

sin((13*π)/8)=−√(,(2+√(,2))/4)

sin((13*π)/8)=−√(,2+√(,2))/2

8. Invert the sine value to find the cosecant:

csc((13*π)/8)=−2/√(,2+√(,2))

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

csc((13*π)/8)=−(2√(,2−√(,2)))/√(,(2+√(,2))*(2−√(,2)))

csc((13*π)/8)=−(2√(,2−√(,2)))/√(,4−2)

csc((13*π)/8)=−(2√(,2−√(,2)))/√(,2)

10. Simplify the expression:

csc((13*π)/8)=−√(,2)⋅√(,2−√(,2))

csc((13*π)/8)=−√(,4−2√(,2))

Final Answer

csc((13*π)/8)=−√(,4−2√(,2))

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