Find the Exact Value csc(210)

Problem

csc(210)

Solution

1. Identify the reference angle by finding the difference between 210 and the nearest x-axis (180.

(θ_ref)=210−180=30

2. Determine the quadrant of the angle 210 which lies in Quadrant III.

3. Determine the sign of the cosecant function in Quadrant III. Since sin(θ) is negative in Quadrant III, its reciprocal csc(θ) is also negative.

csc(210)=−csc(30)

4. Recall the exact value of sin(30) which is 1/2

5. Apply the reciprocal identity csc(θ)=1/sin(θ) to find the value.

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

6. Combine the value with the negative sign from step 3.

csc(210)=−2

Final Answer

csc(210)=−2

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