Find the Exact Value cot((19pi)/6)

Problem

cot((19*π)/6)

Solution

1. Subtract multiples of 2*π to find the coterminal angle within the interval [0,2*π)

(19*π)/6−2*π=(19*π)/6−(12*π)/6

(19*π)/6−(12*π)/6=(7*π)/6

2. Identify the quadrant and the reference angle for (7*π)/6 Since π<(7*π)/6<(3*π)/2 the angle is in Quadrant III.

Reference Angle=(7*π)/6−π

Reference Angle=π/6

3. Determine the sign of the cotangent function in Quadrant III. Cotangent is positive in Quadrant III.

cot((7*π)/6)=cot(π/6)

4. Evaluate the exact value using the known trigonometric values for π/6 (or 30.

cot(π/6)=cos(π/6)/sin(π/6)

cot(π/6)=√(,3)/2/1/2

cot(π/6)=√(,3)

Final Answer

cot((19*π)/6)=√(,3)

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