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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