Solve rho=a(1-cosvarphi)

$$\rho = a ( 1 - \cos \varphi )$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{\rho }{\cos(\phi )-1}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\phi =2\pi n_{1}\\a\in \mathrm{R}\text{, }&\rho =0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\phi =2\pi n_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$a\left(1-\cos(\phi )\right)=\rho$$

Use the distributive property to multiply a by 1-\cos(\phi ).

$$a-a\cos(\phi )=\rho$$

Combine all terms containing a.

$$\left(1-\cos(\phi )\right)a=\rho$$

The equation is in standard form.

$$\left(-\cos(\phi )+1\right)a=\rho$$

Divide both sides by 1-\cos(\phi ).

$$\frac{\left(-\cos(\phi )+1\right)a}{-\cos(\phi )+1}=\frac{\rho }{-\cos(\phi )+1}$$

Dividing by 1-\cos(\phi ) undoes the multiplication by 1-\cos(\phi ).

$$a=\frac{\rho }{-\cos(\phi )+1}$$

Solve for ρ

$$\rho =a\left(-\cos(\phi )+1\right)$$