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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