Find the Exact Value csc(225)

Problem

csc(225^∘)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(225^∘)=1/sin(225^∘)

2. Determine the reference angle for 225^∘ Since 225^∘ is in the third quadrant, subtract 180^∘ from the angle.

Reference Angle=225^∘−180^∘=45^∘

3. Evaluate the sine of the angle. In the third quadrant, the sine function is negative.

sin(225^∘)=−sin(45^∘)

4. Substitute the known value for sin(45^∘) which is √(,2)/2

sin(225^∘)=−√(,2)/2

5. Calculate the cosecant by taking the reciprocal of the sine value.

csc(225^∘)=1/(−√(,2)/2)

6. Simplify the expression by multiplying by the reciprocal and rationalizing the denominator.

csc(225^∘)=−2/√(,2)

csc(225^∘)=−√(,2)

Final Answer

csc(225^∘)=−√(,2)

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