# How to linearise a square root expression?

I have a non-linear ODE that I need to linearise.

I could go about linearising the trig terms using Taylor series...but not sure about a square root of the form:

$$\sqrt{(A+C)^2 + B -2 * \cos (\alpha - \theta)}$$

Thanks.

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Is $\theta$ your variable? – Mike Spivey Dec 9 '10 at 17:59
Could you do a change of variables and make $\sqrt{(A+C)^2+B-2*\cos(\alpha-\theta)}$ a new variable? – Ryan Budney Dec 9 '10 at 21:59
Maybe giving the actual nonlinear ODE might yield answers more useful to you? – J. M. Dec 10 '10 at 0:55

Since you have an ODE, you probably want to be linearizing around an equilbrium point $\theta_0$ (which might not be zero). Note that I'm assuming that $\theta$ is the variable.
Then think about the general form of a Taylor series expansion -- the linearization is given by $$f(\theta) \approx f(\theta_0) + (\theta-\theta_0) f'(\theta_0)$$ where $$f(\theta) = \sqrt{(A+C)^2 + B - 2\cos(\alpha - \theta)}$$
If $(A+C)^2+B$ is large compared with 1 you can divide it out to get $$\sqrt{(A+C)^2+B}\sqrt{1-\frac{2cos(\alpha-\theta)}{(A+C)^2+B}}$$ and use $\sqrt{1-x}\approx 1-\frac{x}{2}$ for $x<<1$. If you are specifically interested in $\theta$ you can expand the cosine term and still use the approximation to the square root considering $\alpha$ to be fixed. Does that help?