Find the Exact Value csc(315 degrees )

Problem

csc(315^∘)

Solution

1. Identify the reference angle for 315^∘ Since 315^∘ is in the fourth quadrant, the reference angle is found by subtracting the angle from 360^∘

(θ_ref)=360^∘−315^∘=45^∘

2. Determine the sign of the cosecant function in the fourth quadrant. In the fourth quadrant, the sine function is negative, and since csc(θ)=1/sin(θ) the cosecant function is also negative.

csc(315^∘)=−csc(45^∘)

3. Recall the exact value of sin(45^∘)

sin(45^∘)=√(,2)/2

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

csc(45^∘)=2/√(,2)

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

2/√(,2)=(2√(,2))/2=√(,2)

6. Apply the negative sign determined in step 2 to find the final value.

csc(315^∘)=−√(,2)

Final Answer

csc(315^∘)=−√(,2)

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