Simplify csc(-120)

Problem

csc(−120^∘)

Solution

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

csc(−120^∘)=−csc(120^∘)

2. Determine the reference angle for 120^∘ Since 120^∘ is in the second quadrant, the reference angle is 180^∘−120^∘=60^∘

csc(120^∘)=csc(60^∘)

3. Evaluate the cosecant of the reference angle. Since csc(θ)=1/sin(θ) and sin(60^∘)=√(,3)/2 we find the reciprocal.

csc(60^∘)=2/√(,3)

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

2/√(,3)=(2√(,3))/3

5. Combine the results by applying the negative sign from the first step.

−csc(120^∘)=−(2√(,3))/3

Final Answer

csc(−120^∘)=−(2√(,3))/3

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