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)

Want more problems? Check here!