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. This approximation is relevant only when angles are measured in radians. Of course, the equality is not exact; only when the angle is zero are the three truly equal. 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.

Mathematical comment

Mathematically, the small angle approximation is the first-order Maclaurin polynomial of the sine function, in the point zero. Recall Maclaurin's theorem:

Let $f:\mathbb {R} \to \mathbb {R}$ 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

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

Which is the result we are familiar with. We can use the approximation here because the approximation becomes better and better as we get closer to the point zero. Choosing an x which is very close to zero is the very point of the limit, so the approximation is valid here.