Solve e=a(1-sinθ)

$$e = a ( 1 - \sin \theta )$$

Solve for a

$$a=-\frac{e}{\sin(\theta )-1}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =2\pi n_{1}+\frac{\pi }{2}$$

Steps for Solving Linear Equation

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

$$a\left(1-\sin(\theta )\right)=e$$

Use the distributive property to multiply a by 1-\sin(\theta ).

$$a-a\sin(\theta )=e$$

Combine all terms containing a.

$$\left(1-\sin(\theta )\right)a=e$$

The equation is in standard form.

$$\left(-\sin(\theta )+1\right)a=e$$

Divide both sides by 1-\sin(\theta ).

$$\frac{\left(-\sin(\theta )+1\right)a}{-\sin(\theta )+1}=\frac{e}{-\sin(\theta )+1}$$

Dividing by 1-\sin(\theta ) undoes the multiplication by 1-\sin(\theta ).

$$a=\frac{e}{-\sin(\theta )+1}$$

Solve for θ

$$\theta =2\pi n_{1}-\arcsin(\frac{a-e}{a})+\pi \text{, }n_{1}\in \mathrm{Z}$$

$$\theta =2\pi n_{2}+\arcsin(\frac{a-e}{a})\text{, }n_{2}\in \mathrm{Z}\text{, }a\geq \frac{e}{2}$$