For angles close to zero, the trigonometric functions can be approximated by simple polynomials. When the angle $\theta$ is measured in radians,

$$\sin\theta=\theta-\frac{\theta^3}{6}+\frac{\theta^5}{120}-\cdots,$$

$$\cos\theta=1-\frac{\theta^2}{2}+\frac{\theta^4}{24}-\cdots.$$

Keeping only the leading terms gives the small-angle approximations

$$\boxed{\sin\theta\approx\theta},$$

$$\boxed{\tan\theta\approx\theta},$$

$$\boxed{\cos\theta\approx1-\frac{\theta^2}{2}}.$$

The approximation $\tan\theta\approx\theta$ follows because $\tan\theta=\sin\theta/\cos\theta$ and $\cos\theta\approx1$ for sufficiently small $|\theta|$.

Why radians matter

These forms are only this simple in radians. If an angle is expressed in degrees, the numerical value of the angle contains an additional conversion factor, so writing $\sin\theta\approx\theta$ with $\theta$ in degrees is incorrect.

For example,

$$5^\circ\approx0.0873,\mathrm{rad}.$$

Then

$$\sin(0.0873)\approx0.0872,$$

which is very close to $0.0873$. By contrast, the statement $\sin 5\approx5$ is meaningless if the $5$ is being interpreted as degrees.

Approximation error

The first omitted term indicates how the error scales. For sine,

$$\sin\theta-\theta\approx-\frac{\theta^3}{6}.$$

Thus halving a sufficiently small angle reduces the leading absolute error by roughly a factor of eight.

At

$$\theta=0.10,\mathrm{rad},$$

$$\sin\theta\approx0.09983,$$

while the approximation gives $0.10$. The relative error is about $0.17%$.

At

$$\theta=0.50,\mathrm{rad},$$

$$\sin\theta\approx0.4794,$$

so replacing it by $0.50$ already gives a relative error of several percent. “Small” therefore depends on the accuracy required by the model.

Why the approximation is useful

Small-angle approximations turn nonlinear geometric relations into linear ones. For example, an equation containing

$$\ddot\theta+\omega_0^2\sin\theta=0$$

becomes, for sufficiently small oscillations,

$$\ddot\theta+\omega_0^2\theta\approx0,$$

which is the harmonic oscillator equation.

The approximation is not a new physical law. It is a controlled local simplification whose validity must be judged from the angle range and the accuracy the problem requires.