# Where is this trig equation derived from?

In the first, top rated, comment on the original post linked below, the author wrote the function $r = \dfrac {\cos(x)} {1−\cos(x)}$. I do not understand how they got to this equation. I get how they solved for $x$ in their first step. Can you please explain how they solved for $r$.

This is the picture of the problem, where we are solving for $r$, given $x = \dfrac {n-2}{2n}$

Original Post Numbers of circles around a circle

• You have linked to a user profile, not a comment or answer. Hence I honestly have no idea what this question is about... – Xander Henderson Sep 11 '18 at 22:06
• I apologize, I am updating the link right now – Levi Kline Sep 12 '18 at 1:09

In the right Triangle, we can write $\cos x$ as Base divided by the hypotenuse

$$\cos x=\frac{r}{r+1}$$

$$\frac{1}{\cos x}=1+\frac{1}{r}$$

$$\frac{1}{r}=\frac{1}{\cos x}-1$$ $$\frac{1}{r}=\frac{1-\cos x}{\cos x}$$ $$r=\frac{\cos x}{1-\cos x }$$

• Where is the 1 from in the denominator at the beginning? Shouldn't it be r/r+L – Levi Kline Sep 11 '18 at 18:41
• it is the length of the hypotenuse in the right triangle which is $r+1$ – Deepesh Meena Sep 11 '18 at 18:42
• it is 1 it is not L – Deepesh Meena Sep 11 '18 at 18:42
• you misreading 1 as L here it is 1 – Deepesh Meena Sep 11 '18 at 18:43
• If my center circle radius is 100, then is it r/r+100? – Levi Kline Sep 11 '18 at 18:43