The small angle approximation

One of the most important results of calculus is the Taylor series expansion. This says that if you know a function only in a small region, but you know its derivatives in that region, then you can use those derivatives to extrapolate beyond that region. You can imagine that this is an important result in scientific applications of mathematics where we rarely know something exactly, but we might have a pretty good idea how it behaves for limited values of a variable. This math lets us make the next step.

One of the results from this math is the description of trig functions for small angles. These results turn out to be of great important in the study of light, both in the theory of lenses, and in the wave theory of interference. But for now, let's just look at the results.

For this analysis, we have to work in radians since radians are dimensionless. As a result all powers of an angle are also unitless and they can be added. 

The result of the advanced math analysis of trig functions of an angle $\theta$ given in radians is that the sine, cosine, and tangent have an expression in terms of powers of the angle:

$$\sin{\theta} = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - ... $$

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

$$\tan{\theta} = \theta + \frac{\theta^3}{3!} + \frac{\theta^5}{5!} + ... $$

where the symbol "N!" means N factorial, equal to the product of all the integers up to and including N:

$$N! = 1 \times 2 \times 3 ... \times N$$.

These are pretty scary! They go on forever (but it's pretty clear what happens next for each case) so what good are they?

Let's just look at the sine first. The first term is just $\theta$. If $\theta$ is small — say 0.1 radians (about 6 degrees) then the second term is $\frac{\theta^3}{6}$ (since 3! = 3 x 2 x 1 = 6) which has a value of 0.00017 = 1.7 x 10^-4. This is pretty small, even compared to 0.1. The next term, $\frac{\theta^5}{120}$ (since 5! = 5 x 4 x 3 x 2 x 1 = 120) only has a value of 0.000000083 = 8.3 x 10^-8. So the first three terms are about 10^-1, 10^-4, and 10^-7.

Unless we want very high accuracy, the first term or two will suffice. (And if you're a mathematician you can even prove that the infinite sum has a finite result and can get an upper bound on the corrections, showing that they are small. This is why some of us really like math!)

So for many applications in physics — where we can keep the angles less than about 20^o, we can use the small angle approximations:

$$\sin{\theta} \approx \theta$$

$$\cos{\theta} \approx 1 - \frac{\theta^2}{2}$$

$$\tan{\theta} \approx \theta$$.

These are the small angle approximations for the trig functions. You will be expected to know and be able to use these. They are important in looking at limiting cases (angles near zero) and for our study of ray optics.

Joe Redish 1/19/14