Find the Exact Value csc(135 degrees )

Problem

csc(135^∘)

Solution

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

csc(135^∘)=1/sin(135^∘)

2. Determine the reference angle for 135^∘ Since 135^∘ is in the second quadrant, the reference angle is 180^∘−135^∘

(θ_ref)=45^∘

3. Evaluate the sine of the angle. In the second quadrant, sine is positive, so sin(135^∘)=sin(45^∘)

sin(135^∘)=√(,2)/2

4. Substitute the value back into the reciprocal identity to find the cosecant.

csc(135^∘)=1/√(,2)/2

5. Simplify the expression by multiplying by the reciprocal.

csc(135^∘)=2/√(,2)

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

csc(135^∘)=√(,2)

Final Answer

csc(135^∘)=√(,2)

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