Find the Exact Value csc(330 degrees )

Problem

csc(330^∘)

Solution

1. Identify the reference angle by finding the difference between 330^∘ and 360^∘ since the angle is in the fourth quadrant.

360^∘−330^∘=30^∘

2. Determine the sign of the cosecant function in the fourth quadrant. Since csc(θ)=1/sin(θ) and sine is negative in the fourth quadrant, the result will be negative.

csc(330^∘)=−csc(30^∘)

3. Apply the reciprocal identity for cosecant.

csc(30^∘)=1/sin(30^∘)

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

csc(30^∘)=1/(1/2)=2

5. Combine the value with the negative sign determined in step 2.

csc(330^∘)=−2

Final Answer

csc(330^∘)=−2

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