Radian

Mark off an arc whose length equals the radius, and the angle it subtends at the center is one radian.

Length over length, so the radian is dimensionless. It is written rad, and it cancels freely: meters per second comes out of $r\omega$ with $r$ in meters and $\omega$ in rad/s, with no leftover unit to explain.

Conversions worth having cold:

• One full revolution is $2\pi$ rad, so $\pi$ rad is 180 degrees.
• One radian is about 57.3 degrees.
• Degrees to radians: multiply by $\pi/180$.

Every rotational equation on the AP sheets assumes radians. The arc relation $s = r\,\Delta\theta$, the linear links $v = r\omega$ and $a_T = r\alpha$, and the three constant-acceleration rotational equations are all derived with the angle in radians. Feed them degrees and the answer is wrong by a factor of $180/\pi$, with no unit mismatch to warn you, because the radian is dimensionless. EK 5.1.A.1 specifies radians for angular displacement for exactly this reason.

Degrees do still appear: the $\sin\theta$ in a torque, the angle of an incline, the direction of a vector. Those are angles between two directions rather than amounts of rotation, and either unit works there as long as your calculator mode matches.

See angular displacement and the AP Physics 1 equation sheet.