Find the Exact Value 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^∘ which is located in the second quadrant.

(θ_ref)=180^∘−120^∘=60^∘

3. Evaluate the sine of 120^∘ using the reference angle, noting that sine is positive in the second quadrant.

sin(120^∘)=sin(60^∘)=√(,3)/2

4. Substitute the value of sin(120^∘) back into the reciprocal identity.

csc(120^∘)=1/√(,3)/2

5. Simplify the expression by multiplying by the reciprocal.

csc(120^∘)=2/√(,3)

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

csc(120^∘)=(2√(,3))/3

Final Answer

csc(120^∘)=(2√(,3))/3

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