mathematics//calculus//Taylor polynomial//small-angle approximation

The small-angle approximation replaces \(\sin\theta\) by \(\theta\), \(\cos\theta\) by 1 and \(\tan\theta\) by \(\theta\) for an angle measured in radians and close to zero, and it is the single most used linearization in engineering: it is what makes a pendulum, a drone near hover or a balancing robot linear enough to design a controller on paper. Each replacement is the first term of a Taylor polynomial at zero, so it is exact at zero and wrong by an amount that grows with the angle.

The small-angle approximation replaces sin⁡θ\sin\thetasinθ by θ\thetaθ, cos⁡θ\cos\thetacosθ by 1 and tan⁡θ\tan\thetatanθ by θ\thetaθ for an angle measured in radians and close to zero, and it is the single most used linearization in engineering: it is what makes a pendulum, a drone near hover or a balancing robot linear enough to design a controller on paper. Each replacement is the first term of a Taylor polynomial at zero, so it is exact at zero and wrong by an amount that grows with the angle.

A planar drone shows what it buys. Its horizontal and vertical accelerations are mx¨=Tsin⁡θm\ddot x=T\sin\thetamx¨=Tsinθ and mz¨=Tcos⁡θ−mgm\ddot z=T\cos\theta-mgmz¨=Tcosθ−mg; at hover the thrust is T∗=mgT^*=mgT∗=mg, and with the approximation they become

x¨≈g θ,z¨≈δTm.\ddot x\approx g\,\theta,\qquad \ddot z\approx \frac{\delta T}{m}.x¨≈gθ,z¨≈mδT​.

Horizontal and vertical motion decouple, and the horizontal one becomes a chain of integrators driven by the tilt: to move sideways the drone must lean, and 5° (0.087 rad) gives about 0.86 m/s per second of acceleration. That linear chain is what LQR or a cascade of PID loops is designed against (planar drone).

The approximation sets the envelope of a hover design.

The replacement sin⁡θ≈θ\sin\theta\approx\thetasinθ≈θ is off by 0.5 % at 10° and nearly 5 % at 30°; cos⁡θ≈1\cos\theta\approx1cosθ≈1 by 1.5 % and 13 %. A controller designed at hover behaves as designed at moderate tilts and degrades in aggressive manoeuvres, where the gains it was tuned with no longer describe the plant.

The cosine fails first. Its error is second order in the angle while the sine's is third order, so at 30° the vertical thrust is 13 % short of what the linear model assumes, and a drone banking hard without compensating sinks. Multirotor autopilots commonly divide the thrust command by the cosine of the tilt for that reason, which is a nonlinear correction layered on a linear design.

The angle must be in radians. In degrees the line sin⁡θ≈θ\sin\theta\approx\thetasinθ≈θ has the wrong slope by a factor of 57, a unit bug that compiles and flies badly.

The same approximation makes a pendulum's period independent of its amplitude, 2πL/g2\pi\sqrt{L/g}2πL/g​, about 2 s for a 1 m pendulum (pendulum). At a swing of 30° the true period is already close to 2 % longer, so a clock or a crane's sway model that assumes small angles drifts when the swing is large.

Past the envelope the options are re-linearizing around the new attitude (gain scheduling), keeping the nonlinear terms in the controller (nonlinear MPC, feedback linearization), or limiting the tilt the vehicle is allowed to command.