Find the Exact Value csc((5pi)/4)

Problem

csc((5*π)/4)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(θ)=1/sin(θ)

2. Determine the reference angle for (5*π)/4 by subtracting π from the angle, since it is in the third quadrant.

(θ_ref)=(5*π)/4−π=π/4

3. Evaluate the sine of the reference angle using known trigonometric values.

sin(π/4)=√(,2)/2

4. Determine the sign of the sine function in the third quadrant, where both sine and cosecant are negative.

sin((5*π)/4)=−√(,2)/2

5. Substitute the sine value back into the reciprocal identity to find the cosecant.

csc((5*π)/4)=1/(−√(,2)/2)

6. Simplify the expression by rationalizing the denominator.

csc((5*π)/4)=−2/√(,2)=−√(,2)

Final Answer

csc((5*π)/4)=−√(,2)

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