Find the Exact Value csc(-240 degrees )

Problem

csc(−240^∘)

Solution

1. Use the odd function property of the cosecant function, which states csc(−θ)=−csc(θ)

csc(−240^∘)=−csc(240^∘)

2. Find the reference angle for 240^∘ Since 240^∘ is in the third quadrant, the reference angle is 240^∘−180^∘=60^∘

Reference Angle=60^∘

3. Determine the sign of the cosecant function in the third quadrant. Since sine (and therefore cosecant) is negative in the third quadrant, csc(240^∘)=−csc(60^∘)

−csc(240^∘)=−(−csc(60^∘))

4. Simplify the signs to show that the expression is positive.

−(−csc(60^∘))=csc(60^∘)

5. Evaluate the cosecant of the reference angle. Since sin(60^∘)=√(,3)/2 its reciprocal is csc(60^∘)=2/√(,3)

csc(60^∘)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

csc(−240^∘)=(2√(,3))/3

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