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According to wolfram alpha, $\dfrac {\sin \theta} {1 - \cos \theta} = \cot \left(\dfrac{\theta}{2} \right)$.

But how would you get to $\cot \left(\dfrac{\theta}{2} \right)$ if you're given $\dfrac {\sin \theta} {1 - \cos \theta}$?

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4 Answers 4

up vote 9 down vote accepted

All you need are the following formulas:

$$\sin(2A) = 2 \sin(A) \cos(A) \text{ and } \cos(2A) = 1 - 2 \sin^2(A)$$ Now take $A = \theta/2$ to get that $$\sin(\theta) = 2 \sin(\theta/2) \cos(\theta/2) \text{ and } \cos(\theta) = 1 - 2 \sin^2(\theta/2)$$ Now plugging this into your equation, we get that $$\dfrac{\sin(\theta)}{1-\cos(\theta)} = \dfrac{2 \sin(\theta/2) \cos(\theta/2)}{2 \sin^2(\theta/2)} = \dfrac{\cos(\theta/2)}{\sin(\theta/2)} = \cot(\theta/2)$$

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Oh yeah, thanks. Seems obvious now. –  Cameron Martin Jun 20 '12 at 21:24

Proof without words: $\displaystyle\cot(\theta/2)=\frac{\color{green}{\sin(\theta)}}{\color{red}{1-\cos(\theta)}}$

$\hspace{3.5cm}$enter image description here

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1  
+1. I remember seeing a similar pic somewhere on this website which was also posted by you. Am I right or am I dreaming? –  user17762 Jun 20 '12 at 22:22
    
@Marvis: I don't know what you mean ;-) –  robjohn Jun 20 '12 at 22:25
    
Ya. Right that was the pic! Good to know that I am not imagining things :). –  user17762 Jun 20 '12 at 22:34
    
To nitpick, you need a different picture for $\theta \notin [-\pi/2, \pi/2]$ –  user17762 Jun 20 '12 at 22:36
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@Marvis: actually, I think this still works. –  robjohn Jun 20 '12 at 23:08

Alternately, use these formulas: $$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2} \mbox{and} \, \sin(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i} \, .$$

So \begin{eqnarray*} \frac{\sin(\theta)}{1-\cos(\theta)} & = & \frac{(e^{i \theta} - e^{-i \theta})/(2i)}{(2-e^{i \theta}-e^{-i\theta})/2}\\ & = & \frac{1}{i}\frac{(e^{i \theta/2} +e^{-i\theta/2})(e^{i\theta/2}-e^{-i\theta/2})}{-(e^{i \theta/2}-e^{-i\theta/2})^2}\\ & = & i\frac{e^{i \theta/2}+e^{-i\theta/2}}{e^{i\theta/2}-e^{-i\theta/2}}\\ & =& i \frac{2\cos (\theta/2)}{2i \sin (\theta/2)}\\ & =& \cot(\theta/2) \, .\end{eqnarray*}

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There is a very standard way of transforming trigonometric functions into rational functions of a different variable - generally first learned when integrating functions, but heavily related to the unit circle and illustrative of basic concepts in algebraic geometry.

One sets $t=\tan \frac \theta 2$, when $\sin \theta = \frac {2t} {1+t^2} $ and $\cos \theta = \frac {1-t^2}{1+t^2}$

This gives the result easily, and is worth knowing.

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For searching purposes: this is the Weierstrass substitution. –  J. M. Jul 10 '12 at 14:00

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