# Simplifying an equation when finding the potential energy of a wave

I'm sorry if this question is not at the right place, I didn't know if I should post it in the math or physics stackexchange...
Anyways, I encountered a step in a derivation that I can't really figure out. I'm trying to figure out the potential energy of a wave on a string and it starts like this:

To a good approximation, the extended length of a string segment $$\delta s$$ is related to the unstretched length $$\delta x$$

$$\delta s = \frac{\delta x}{\cos{\theta}} = \frac{\delta x}{(1-\sin^2\theta)^{1/2}}$$
Since $$\theta$$ is small,

$$\delta s \simeq \frac{\delta x}{(1-\theta^2)^{1/2}} \simeq \delta x (1+\frac{1}{2}\theta^2)$$
This last step is the thing I don't get... Is it some kind of authorized approximation from the small angle? I don't see where the square root goes or how the one half pops out, but this is what is witten in my text book...
Also sorry hat I bring small angle approximations to you, the mathematicians, I've heard you are not very big fans of them ;) I thank you in advance for your help!

## 1 Answer

We have that by Binomial approximation for $$x$$ small

$$(1+x)^r \approx 1+rx$$

here $$r=-\frac12$$ and then

$$\delta s \simeq \frac{\delta x}{(1-\theta^2)^{1/2}} = \delta x(1-\theta^2)^{-1/2} \simeq\delta x \left(1+\frac{1}{2}\theta^2\right)$$