Arctan2 vs arctan

Author:
DavidK

Motivation

Given a point (x,y) in the rectangular xy-plane, what angle, theta, does the line segment from the origin to (x,y) and the positive x-axis form? Or in other words, in which direction does the vector u point? This question seems to be answered by using the arctan function. atan(y/x). However, this is only true for some points (x,y). Move the point in the following graph to find where this is true.

When does arctan(y/x) give the angle theta? When point (x,y) is in ...

Check all that apply
The angle theta and atan(b/a) are equal when the point (a,b) is in the first quadrant. In the IV quadrant atan(b/a) is negative. Having a negative angle signifies that the point is below the x axis, which is fine. Notice -45=315. When (a,b) is in the II or III quadrant arctan provides an angle that is 180 degrees off. This is because a/b=-a/-b and -a/b=a/-b. Now compare the angle theta with arctan2.

arctan2 finds the angle

The function arctan2(y,x) finds the angle direction of the point (x,y) perfectly. We do not need to compensate for when x is negative (|theta|>90). The range of arctan2 is (-180,180].