Which quadrant is (4,11π6) in?

1 Answer
Nov 24, 2015

Quadrant III

Explanation:

To find a point (r,θ) in polar coordinates, imagine an arrow which rotates θ 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 3π2 is a three-quarter rotation,2π is a full rotation, and 3π2<11π6<2π
rotating 11π6 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(θ) and y=rsin(θ)

Then here,
x=(4)cos(11π6)=23
y=(4)sin(11π6)=2

and (23,2) is in quadrant III