Small-angle approximation

AP Physics C: Mechanics writes it out at EK 7.5.A.2.ii: for small amplitudes of motion, the small-angle approximation can be applied to the restoring torque, giving $\sin\theta \approx \theta$ and $\tau = -mgd\theta = I\alpha$. EK 7.5.A.2.iii then says the small-angle approximation and Newton's second law in rotational form yield a second-order differential equation that describes SHM, $\frac{d^2\theta}{dt^2} = -\omega^2\theta$.

Radians, and only radians. The two sides are not close in degrees: $\sin 10^\circ = 0.174$ while the number 10 is not. In radians, $10^\circ$ is $0.175$ rad against $\sin\theta = 0.174$, and the approximation overstates $\sin\theta$ by about 0.5 percent. At $30^\circ$ the gap is about 4.7 percent. This is the clearest reason the radian is the rotational unit rather than a preference.

It is the fine print on the pendulum period. AP Physics 1 states the restriction inside the essential knowledge rather than in a boundary statement. EK 7.2.A.1.ii introduces $T_p = 2\pi\sqrt{\ell/g}$ as the period of a simple pendulum displaced by a small angle, and EK 7.1.A.2.iii says the motion of a pendulum with a small angular displacement can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement. Swing further and the proportionality fails first, the SHM label goes with it, and the formula follows.

AP Physics 2 is where a number appears. EK 14.8.A.1.v reads: for small angles, where $\theta < 10^\circ$, the small angle approximation can be used to relate $\lambda$, $d$ and $L$ to $y_{\text{max}}$. The Physics 2 exam conventions box adds a standing assumption, that the small angle approximation is valid for single- and double-slit diffraction.