Solve a=r(1-sinθ)

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

Solve for r

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

Steps for Solving Linear Equation

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

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

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

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

Combine all terms containing r.

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

The equation is in standard form.

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

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

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

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

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

Solve for a

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