Simplify csc(225)

Problem

csc(225^∘)

Solution

1. Identify the reference angle by determining the quadrant of 225^∘ Since 180^∘<225^∘<270^∘ the angle is in the third quadrant.

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 the third quadrant. Since sin(θ) is negative in the third quadrant, csc(θ) is also negative.

csc(225^∘)=−csc(45^∘)

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

csc(45^∘)=1/sin(45^∘)

5. Substitute the known value sin(45^∘)=√(,2)/2

csc(45^∘)=1/√(,2)/2=2/√(,2)

6. Rationalize the denominator.

2/√(,2)⋅√(,2)/√(,2)=(2√(,2))/2=√(,2)

7. Combine the value with the negative sign determined in step 3.

csc(225^∘)=−√(,2)

Final Answer

csc(225^∘)=−√(,2)

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