Which quadrant is (-4, -11pi/6) in?

1 Answer
Nov 24, 2015

Quadrant III

Explanation:

To find a point (r, theta) in polar coordinates, imagine an arrow which rotates theta radians counterclockwise from the pole (what would be the positive x axis in rectangular coordinates) and then extends r in that direction.

Rotating negative radians counterclockwise is the same as rotating the same number clockwise. Thus, as 3pi/2 is a three-quarter rotation,2pi is a full rotation, and 3pi/2 < (11pi)/6 < 2pi
rotating (11pi)/6 in the clockwise direction will result in pointing to quadrant I

However, similarly, extending a negative value is the same as extending a positive value in the opposite direction. Thus the resulting arrow will be in quadrant III


Alternatively, you could note that to convert from polar coordinates to rectangular, we let x = rcos(theta) and y = rsin(theta)

Then here,
x = (-4)cos(-(11pi)/6) = -2sqrt(3)
y = (-4)sin(-(11pi)/6) = -2

and (-2sqrt(3), -2) is in quadrant III