# Simplify equation

1. Feb 1, 2014

### hermano

1. The problem statement, all variables and given/known data

In a paper I have found there is a simplification of an equation which I do not understand.
How did they do this? Can someone explain this step by step?

2. Relevant equations
The euqation is as follows:

X = $e \cos\theta + \sqrt{(R^2 + e^2 \sin^2 \theta)}$

X = distance
e = eccentricity
theta = angle

Since the eccentricity is much than 1000 times smaller than the radius the above equation is simplified to:

X = $e \cos\theta + R + \frac{e^2}{2R} \sin^2 \theta$

3. The attempt at a solution
In the paper they say that they used the Taylor series expansion in terms of $\sin^2 \theta$ for this simplification.

Last edited: Feb 1, 2014
2. Feb 1, 2014

### Staff: Mentor

The Taylor expansion being made is that of $(1+x)^{1/2}$ with $|x| \ll 1$:
\begin{align} (R^2 + e^2 \sin^2 \theta)^{1/2} &= R \left( 1 + \frac{e^2}{R^2} \sin^2 \theta \right)^{1/2} \\ &\approx R \left( 1 + \frac{e^2}{2 R^2} \sin^2 \theta \right) \\ &= R + \frac{e^2}{2 R} \sin^2 \theta \end{align}

3. Feb 2, 2014

### hermano

Thank you DrClaude. Can you also help me to simplify the following equation using the Taylor series:

$(R^2-e^2 \sin^2 \theta)^{1/2}$

4. Feb 2, 2014

### Staff: Mentor

Follow the example DrClaude has given, except now replace every x by (-x) in the Taylor expansion:

(1+x)^n = 1 + (n)(x) + (n)(n-1)(x^2)/2! + ...

5. Feb 2, 2014

### hermano

f(x) = (1-x)^(1/2) -> f(0)= 1
f'(x) = (1/2)(1-x)^(-1/2) -> f'(0) = 1/2

So the Tyalor expansion will become:
(1-x)^(1/2) = 1 + (1/2) (x)

and not
(1-x)^(1/2) = 1 - (1/2) (x) as when I replace x by (-x) in the Taylor expansion for (1+x)^(1/2)

Or what do I wrong?

6. Feb 2, 2014

### Staff: Mentor

There is a minus sign missing:
\begin{align} \frac{d}{dx} (1-x)^{1/2} &= \frac{1}{2} (1-x)^{-1/2} \frac{d}{dx} (1-x) \\ &= -\frac{1}{2} (1-x)^{-1/2} \end{align}

7. Feb 3, 2014

### hermano

Thanks! That's the error I have made.