Find the Exact Value csc(225 degrees )

Problem

csc(225)

Solution

1. Identify the reference angle for 225 by determining its position in the coordinate plane. Since 180<225<270 the angle is in Quadrant III.

2. Calculate the reference angle θ′ using the formula θ′=θ−180

θ′=225−180=45

3. Determine the sign of the cosecant function in Quadrant III. In this quadrant, both sine and cosecant are negative.

4. Apply the reciprocal identity csc(θ)=1/sin(θ)

csc(225)=−csc(45)

5. Substitute the known value for sin(45)=√(,2)/2 which means csc(45)=2/√(,2)

csc(225)=−2/√(,2)

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

−(2√(,2))/2=−√(,2)

Final Answer

csc(225)=−√(,2)

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