Solve Dr=a(1-sinθ)

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

Solve for D

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

Steps for Solving Linear Equation

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

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

The equation is in standard form.

$$rD=-a\sin(\theta )+a$$

Divide both sides by r.

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

Dividing by r undoes the multiplication by r.

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

Solve for a

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

Steps for Solving Linear Equation

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

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

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

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

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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