Simplify csc(120)

Problem

csc(120^∘)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(120^∘)=1/sin(120^∘)

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

sin(120^∘)=sin(60^∘)

3. Substitute the known exact value for sin(60^∘) which is √(,3)/2

sin(120^∘)=√(,3)/2

4. Calculate the reciprocal of the sine value to find the cosecant.

csc(120^∘)=1/√(,3)/2

5. Simplify the fraction by multiplying by the reciprocal.

csc(120^∘)=2/√(,3)

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

csc(120^∘)=(2√(,3))/3

Final Answer

csc(120^∘)=(2√(,3))/3

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