Find the Exact Value csc((7pi)/6)

Problem

csc((7*π)/6)

Solution

1. Identify the reference angle by determining how far the angle is from the nearest xaxis.

(7*π)/6=π+π/6

Reference Angle=π/6

2. Determine the quadrant of the angle (7*π)/6 Since it is between π and (3*π)/2 the angle lies in Quadrant III.

3. Determine the sign of the cosecant function in Quadrant III. Since csc(θ)=1/sin(θ) and sine is negative in Quadrant III, cosecant is also negative.

4. Apply the reference angle and the sign to the cosecant function.

csc((7*π)/6)=−csc(π/6)

5. Evaluate the cosecant of the reference angle using the reciprocal of sin(π/6)=1/2

csc(π/6)=1/(1/2)=2

6. Combine the value with the negative sign.

csc((7*π)/6)=−2

Final Answer

csc((7*π)/6)=−2

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