Find the Exact Value csc(240)

Problem

csc(240)

Solution

1. Identify the reference angle by finding the distance from the angle to the nearest x-axis. Since 240 is in the third quadrant, the reference angle is 240−180=60

2. Determine the sign of the cosecant function in the third quadrant. In the third quadrant, the sine function is negative, which means its reciprocal, csc(θ) is also negative.

3. Evaluate the cosecant of the reference angle. We know that sin(60)=√(,3)/2 so its reciprocal is csc(60)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

5. Combine the value with the correct sign from step 2.

csc(240)=−(2√(,3))/3

Final Answer

csc(240)=−(2√(,3))/3

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