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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