Find the Exact Value csc(240 degrees )

Problem

csc(240^∘)

Solution

1. Identify the reference angle for 240^∘ Since 240^∘ is in the third quadrant, the reference angle is calculated as 240^∘−180^∘=60^∘

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

3. Apply the reference angle and the sign to the cosecant function.

csc(240^∘)=−csc(60^∘)

4. Recall the value of sin(60^∘) which is √(,3)/2

5. Calculate the reciprocal to find csc(60^∘)

csc(60^∘)=1/sin(60^∘)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

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

csc(240^∘)=−(2√(,3))/3

Final Answer

csc(240^∘)=−(2√(,3))/3

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