Find the Exact Value csc(660)

Problem

csc(660)

Solution

1. Find the coterminal angle by subtracting 360 from 660 to bring the angle within the standard 0 to 360 range.

660−360=300

2. Identify the quadrant for 300 Since 270<300<360 the angle lies in Quadrant IV.

3. Determine the reference angle for an angle in Quadrant IV by subtracting the angle from 360

(θ_ref)=360−300=60

4. Apply the reciprocal identity for cosecant, which is the reciprocal of the sine function.

csc(300)=1/sin(300)

5. Determine the sign of the sine function in Quadrant IV. Since sine is negative in Quadrant IV, the value will be negative.

sin(300)=−sin(60)

6. Substitute the known value for sin(60) which is √(,3)/2

sin(300)=−√(,3)/2

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

csc(300)=1/(−√(,3)/2)=−2/√(,3)

8. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

−2/√(,3)⋅√(,3)/√(,3)=−(2√(,3))/3

Final Answer

csc(660)=−(2√(,3))/3

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