Approximation of the equation of ellipse

[Leo Liu]

TL;DR Summary

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My physics textbook does the approximation that $$r=\frac{r_0}{1-\frac{A}{r_0}\sin\theta}\approx r_0\left( 1+\frac A r_0\sin\theta\right)$$ when $A/r_0 \ll 1$. Can someone please explain how it is done?

 

[fresh_42]

$(1-x)^{-1}=\sum_{k=0}^\infty x^k \approx 1+x$ if $|x|\ll 1.$ The terms $x^2,x^3,$ etc. are neglected.

 

[Office_Shredder]

For what it's worth, basically every approximation you see like this in physics comes from the Taylor series of a function. The fact that you see an $\frac{A}{r_0}\sin(\theta)$ suggests that is what's being plugged into the function.