Find the Exact Value csc(-210 degrees )

Problem

csc(−210)

Solution

1. Use the odd function property of the cosecant function, which states csc(−θ)=−csc(θ)

csc(−210)=−csc(210)

2. Find the reference angle for 210 Since 210 is in the third quadrant, the reference angle is 210−180=30

Reference Angle=30

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

csc(210)=−csc(30)

4. Substitute this back into the expression from step 1.

−csc(210)=−(−csc(30))

csc(−210)=csc(30)

5. Evaluate the cosecant of the reference angle. Since sin(30)=1/2 its reciprocal is 2

csc(30)=2

Final Answer

csc(−210)=2

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