Solve varphi=a(1+cosθ)

$$\varphi = a ( 1 + \cos \theta )$$

Solve for a

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

Steps for Solving Linear Equation

Use the distributive property to multiply a by 1+\cos(\theta ).

$$\phi =a+a\cos(\theta )$$

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

$$a+a\cos(\theta )=\phi$$

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by 1+\cos(\theta ).

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

Dividing by 1+\cos(\theta ) undoes the multiplication by 1+\cos(\theta ).

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