Find the Exact Value csc(-120 degrees )

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. Identify the reference angle for 120 Since 120 is in the second quadrant, the reference angle is 180−120=60

csc(120)=csc(60)

3. Determine the sign of the cosecant function in the second quadrant. Since sin(θ) is positive in the second quadrant, csc(θ) is also positive.

−csc(120)=−csc(60)

4. Evaluate the cosecant of the reference angle. Since sin(60)=√(,3)/2 the cosecant is the reciprocal.

csc(60)=2/√(,3)

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

2/√(,3)=(2√(,3))/3

6. Combine the results to find the final value.

−csc(60)=−(2√(,3))/3

Final Answer

csc(−120)=−(2√(,3))/3

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