Find the Exact Value csc(135)

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^∘ which is located in the second quadrant.

(θ_ref)=180^∘−135^∘=45^∘

3. Evaluate the sine of 135^∘ using the reference angle and the fact that sine is positive in the second quadrant.

sin(135^∘)=sin(45^∘)=√(,2)/2

4. Substitute the value of sin(135^∘) back into the reciprocal identity.

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√(,2))/2=√(,2)

Final Answer

csc(135^∘)=√(,2)

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