Find the Exact Value csc((11pi)/6)

Problem

csc((11*π)/6)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(θ)=1/sin(θ)

2. Determine the reference angle for (11*π)/6 by finding its distance from the nearest x-axis, which is 2*π

Reference Angle=2*π−(11*π)/6

Reference Angle=π/6

3. Determine the quadrant of the angle. Since (11*π)/6 is between (3*π)/2 and 2*π it lies in Quadrant IV.

4. Evaluate the sine of the angle. In Quadrant IV, sine is negative.

sin((11*π)/6)=−sin(π/6)

sin((11*π)/6)=−1/2

5. Substitute the sine value back into the reciprocal identity to find the cosecant.

csc((11*π)/6)=1/(−1/2)

csc((11*π)/6)=−2

Final Answer

csc((11*π)/6)=−2

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