Find the Exact Value csc(-135)

Problem

csc(−135)

Solution

1. Apply the odd function property of the cosecant function, which states csc(−θ)=−csc(θ)

csc(−135)=−csc(135)

2. Find the reference angle for 135 Since 135 is in the second quadrant, the reference angle is 180−135=45

(θ_ref)=45

3. Determine the sign of the cosecant function in the second quadrant. Since sin(θ) is positive in the second quadrant, csc(θ) is also positive.

csc(135)=csc(45)

4. Evaluate the cosecant of the reference angle. Since sin(45)=√(,2)/2 its reciprocal is csc(45)=√(,2)

csc(45)=√(,2)

5. Combine the results by substituting the value back into the expression from step 1.

−csc(135)=−√(,2)

Final Answer

csc(−135)=−√(,2)

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