Find the Quadrant of the Angle (19pi)/6

Problem

(19*π)/6

Solution

1. Identify the angle in radians and determine how many full rotations (2*π it contains to find a coterminal angle within the range [0,2*π)

2. Divide the numerator by the denominator to express the fraction as a mixed number.

(19*π)/6=3*π+π/6

3. Subtract the largest possible multiple of 2*π to find the coterminal angle θ

θ=(19*π)/6−2*π

θ=(19*π)/6−(12*π)/6

θ=(7*π)/6

4. Compare the coterminal angle θ=(7*π)/6 to the standard quadrant boundaries in radians: Quadrant I (0,π/2) Quadrant II (π/2,π) Quadrant III (π,(3*π)/2) and Quadrant IV ((3*π)/2,2*π)

5. Determine the location since π<(7*π)/6<(3*π)/2

Final Answer

Quadrant of (19*π)/6=Quadrant III

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