Find the Exact Value csc(-150)

Problem

csc(−150)

Solution

1. Apply the odd function identity for the cosecant function, which states that csc(−θ)=−csc(θ)

csc(−150)=−csc(150)

2. Find the reference angle for 150 Since 150 is in the second quadrant, the reference angle is 180−150=30

(θ_ref)=30

3. Determine the sign of the cosecant function in the second quadrant. Since sin(θ) is positive in the second quadrant, csc(θ) is also positive.

csc(150)=csc(30)

4. Evaluate the cosecant of the reference angle using the reciprocal identity csc(θ)=1/sin(θ)

csc(30)=1/sin(30)

5. Substitute the known value sin(30)=1/2 into the expression.

csc(30)=1/(1/2)=2

6. Combine the results by applying the negative sign from the first step.

−csc(150)=−2

Final Answer

csc(−150)=−2

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