# How did they “divide the equations”?

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

Two 5.0g point charges on 1.0m-long threads repel each other after being charged to +100nC.
What is angle theta? You can assume theta is a small angle.

2. Relevant equations
$K=9.0*10^9 Nm^2/C^2$
$g=9.8m/s^2$
$q=\frac{K|q_1||q_2|}{d^2}$

3. The attempt at a solution

I determined my unknowns and translated everything into standard units, and drew out the free-body diagram.

Translated everything into standard units.
5.0g = 5.0*10^(-3) kg
100nc = 100*10^(-9) C

$\frac{1}{2}d=\sin{\theta}$
$d=2\sin{\theta}$
$T_x=T\sin{\theta}=F_{2on1}=\frac{Kq^2}{d^2}$
$T_x=T\sin{\theta}=F_{2on1}=\frac{Kq^2}{(2\sin{\theta})^2}$
$T_x=T\sin{\theta}=F_{2on1}=\frac{Kq^2}{4sin^2{\theta} }$
$T_y=T\cos{\theta}=mg$

The "missing" step:
The solutions manual says that the next step from here is to "divide the two equations and solve for q". What I don’t know is how they "divided the two equations" to get this:

$\sin^2{\theta}\tan{\theta}=\frac{Kq^2}{4L^2mg}=4.59*10^{-4}$

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