Find the Exact Value csc(120 degrees )

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 and the fact 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 and rationalizing the denominator.

csc(120)=2/√(,3)

csc(120)=(2√(,3))/3

Final Answer

csc(120)=(2√(,3))/3

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