Find the Exact Value csc(150 degrees )

Problem

csc(150^∘)

Solution

1. Identify the reference angle for 150^∘ Since 150^∘ is in the second quadrant, the reference angle is calculated as 180^∘−150^∘=30^∘

2. Determine the sign of the cosecant function in the second quadrant. In the second quadrant, the sine function is positive, and since csc(θ)=1/sin(θ) the cosecant function is also positive.

3. Apply the reference angle property.

csc(150^∘)=csc(30^∘)

4. Evaluate the cosecant of the reference angle using the reciprocal identity.

csc(30^∘)=1/sin(30^∘)

5. Substitute the known value sin(30^∘)=1/2

csc(30^∘)=1/(1/2)=2

Final Answer

csc(150^∘)=2

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