Small angle approximation

The small angle approximation is a rule that says that for small angles, the trigonometric functions sine and tangent are approximately equal to the angle. The approximation is valid only when angles are measured in radians. Of course, the equality is not exact except when the angle is zero. In symbolic terms:

$\theta \approx \sin \theta \approx \tan \theta \!$

Using the rule, a physical equation such as the equation for diffraction minima:

$b\sin(\theta )=m\lambda \!$

might become

$b\theta =m\lambda \!$.

The equation can then be written as a pure product of quantities with whole-numbered exponents, which is sometimes useful.

The rule is very useful to an engineer performing experiments or making approximations. It is probably best to avoid it when accuracy is important, or when larger angles are expected.

Pendulum motion

The small angle approximation is used to solve for the motion of a pendulum when studying oscillatory motion in introductory physics courses. As long as the pendulum string makes a small angle from the vertical, the approximation is valid and simplifies solving the equation of motion for the pendulum.

Mathematical comment

Mathematically, the small angle approximation is the first-order Maclaurin series of the sine function about the value zero. Recall Maclaurin's theorem:

Let $f:R\to R$ be a function which is n times differentiable in some proximity of the point zero. The Maclaurin polynomial order n is defined as:

$f(0)+x{\frac {f'(0)}{1!}}+x^{2}{\frac {f''(0)}{2!}}+...+x^{n}{\frac {f^{(n)}(0)}{n!}}+({\textrm {remainder}}),$

where the remainder approaches zero as $x\to 0$ and as $n\to \infty$. We find the first-order Maclaurin approximation as

$\sin(0)+x\cos(0)=x.\!$

This shows how the small angle approximation is arrived at. The approximation can be used for purely mathematical purposes as well:

Say that we want to find the limit

$\lim _{x\to 0}\quad {\frac {\sin x}{x}}.$

Substituting the approximation for sine, we get

$\lim _{x\to 0}\quad {\frac {x}{x}}=1.$

We can use the approximation here because the approximation becomes better and better as x approaches zero.