Find the Exact Value csc(-315 degrees )

Problem

csc(−315)

Solution

1. Use the property of odd functions to rewrite the expression, since csc(−θ)=−csc(θ)

csc(−315)=−csc(315)

2. Find the reference angle for 315 which is in the fourth quadrant.

(θ_ref)=360−315=45

3. Determine the sign of the cosecant function in the fourth quadrant. Since sine is negative in the fourth quadrant, cosecant is also negative.

csc(315)=−csc(45)

4. Substitute this back into the expression from step 1.

−csc(315)=−(−csc(45))=csc(45)

5. Evaluate the cosecant of 45 using the reciprocal identity csc(θ)=1/sin(θ)

csc(45)=1/sin(45)=1/√(,2)/2=√(,2)

Final Answer

csc(−315)=√(,2)

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