Find the Exact Value csc(-(5pi)/6)

Problem

csc(−(5*π)/6)

Solution

1. Apply the odd function property of the cosecant function, which states that csc(−x)=−csc(x)

csc(−(5*π)/6)=−csc((5*π)/6)

2. Identify the reference angle for (5*π)/6 Since the angle is in the second quadrant, the reference angle is π−(5*π)/6=π/6

Reference Angle=π/6

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

csc((5*π)/6)=csc(π/6)

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

csc(π/6)=1/sin(π/6)

5. Substitute the known value sin(π/6)=1/2 into the expression.

csc(π/6)=1/(1/2)=2

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

−csc((5*π)/6)=−2

Final Answer

csc(−(5*π)/6)=−2

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