Find the Exact Value csc(315)

Problem

csc(315^∘)

Solution

1. Identify the reference angle by finding the difference between 315^∘ and 360^∘ since the angle is in the fourth quadrant.

360^∘−315^∘=45^∘

2. Determine the sign of the cosecant function in the fourth quadrant. Since csc(θ)=1/sin(θ) and sine is negative in the fourth quadrant, the result will be negative.

csc(315^∘)=−csc(45^∘)

3. Apply the reciprocal identity for cosecant, which states that csc(θ)=1/sin(θ)

csc(45^∘)=1/sin(45^∘)

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

csc(45^∘)=1/√(,2)/2

5. Simplify the fraction by taking the reciprocal.

csc(45^∘)=2/√(,2)

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

(2√(,2))/2=√(,2)

7. Combine the value with the negative sign determined in step 2.

csc(315^∘)=−√(,2)

Final Answer

csc(315^∘)=−√(,2)

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