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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