Angular displacement

How far has it turned, and which way? Angular displacement is that answer, in radians. EK 5.1.A.1: it is the measurement of the angle, in radians, through which a point on a rigid system rotates about a specified axis. The symbol is $\Delta\theta$ and the unit is the radian.

Choose a positive direction first. EK 5.1.A.1.ii says one direction of rotation about the axis, clockwise or counterclockwise, is typically taken as mathematically positive, with the other becoming negative. Which one is yours to pick; counterclockwise positive is the usual choice. Declare it, then hold it for the whole problem.

AP Physics 1 keeps rotation flat. The Topic 5.1 boundary statement limits descriptions of the direction of rotation to clockwise and counterclockwise with respect to a given axis. There is no right-hand rule and no angular displacement vector in this course.

Angular displacement connects to linear distance through the radius: a point a distance $r$ from the axis sweeps an arc

and that only holds with $\Delta\theta$ in radians.

It is not a count of turns. Three and a half rotations is $7\pi$ rad. A wheel that turns forward two revolutions and back two has zero angular displacement even though it moved the whole time, the same way linear displacement differs from distance traveled.

The three constant-acceleration rotational equations are handled in rotational kinematics.