Simplify csc(150)

Problem

csc(150^∘)

Solution

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

csc(150^∘)=1/sin(150^∘)

2. Determine the reference angle for 150^∘ by subtracting it from 180^∘ since it is in the second quadrant.

(θ_ref)=180^∘−150^∘=30^∘

3. Evaluate the sine of the angle, noting that sine is positive in the second quadrant.

sin(150^∘)=sin(30^∘)

4. Substitute the known value of sin(30^∘)=1/2 into the expression.

sin(150^∘)=1/2

5. Calculate the reciprocal to find the final value of the cosecant function.

csc(150^∘)=1/(1/2)=2

Final Answer

csc(150^∘)=2

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