2
$\begingroup$

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the Taylor series for $\arctan \left( \frac{\tan \theta}{2} \right)$. Sure enough, the first three terms are $0 + \frac12 \theta + 0 \theta^2$.

But what I didn't expect was the striking pattern in the next few terms: $$ \arctan \left( \frac{\tan \theta}{2} \right) = \frac12 \theta + \frac{1}{8} \theta^3 + \frac{1}{32} \theta^5 + \frac{11}{1920} \theta^7 + \cdots $$ It clearly seems to be the case that \begin{align*} \arctan \left( \frac{\tan \theta}{2} \right) &\approx \sum_{n \ge 0} \left(\frac{\theta}{2}\right)^{2n+1} \\ &= \frac{\theta / 2}{1 - \theta^2 / 4} \\ &= \frac{1}{2 - \theta} - \frac{1}{2 + \theta}. \\ \end{align*} Here is a plot:

enter image description here

Honestly, I have no idea how to explain this, and in particular I'm having trouble assigning any meaning to $\frac{1}{2 - x}$ at all when $x$ is an angle.

So, my question: is there any intuitive geometric reason why this approximation works? If not, is there any other concise reason?

Obviously one can arrive at this result via algebraic manipulation (like I did above) but I am interested in a more intuitive explanation.

$\endgroup$
  • $\begingroup$ Go one more step, Neglect $ \theta^2 <<1 $ and you get there. $\endgroup$ – Narasimham Mar 1 '18 at 21:25
7
$\begingroup$

The approximation you noticed is the Padé approximant of order $(m,n) = (1,2)$ of $f(\theta) = \tan^{-1} \bigl( \frac{1}{2} \tan \theta \bigr)$.

$\endgroup$
  • $\begingroup$ "Geometric" in what sense, precisely? You need to clearly explain what you mean. You utilized series expansions as if that were a perfectly natural and intuitive thing to do, yet you don't ask for a geometric explanation for those. A Padé approximant is simply a generalization of series expansions to rational functions. If you want "intuitive" explanations, you have to establish what you would characterize as intuitive and what you would not. $\endgroup$ – heropup May 11 '14 at 0:38
  • $\begingroup$ Geometric means arising from a geometric picture. For instance there are very straightforward proofs that $\sin x / x$ tends to $1$ or that the area of a circle is $\pi r^2$ using geometry. And "intuitive" means you understand why it is, not just how to prove it. $\endgroup$ – 6005 May 11 '14 at 0:41
  • $\begingroup$ Okay, that's fair enough. I will leave it to others to do that. $\endgroup$ – heropup May 11 '14 at 0:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.