Solve r=a(999-sinθ)

$$r = a ( 999 - \sin \theta )$$

Solve for a

$$a=-\frac{r}{\sin(\theta )-999}$$

Steps for Solving Linear Equation

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

$$a\left(999-\sin(\theta )\right)=r$$

Use the distributive property to multiply a by 999-\sin(\theta ).

$$999a-a\sin(\theta )=r$$

Combine all terms containing a.

$$\left(999-\sin(\theta )\right)a=r$$

The equation is in standard form.

$$\left(-\sin(\theta )+999\right)a=r$$

Divide both sides by 999-\sin(\theta ).

$$\frac{\left(-\sin(\theta )+999\right)a}{-\sin(\theta )+999}=\frac{r}{-\sin(\theta )+999}$$

Dividing by 999-\sin(\theta ) undoes the multiplication by 999-\sin(\theta ).

$$a=\frac{r}{-\sin(\theta )+999}$$

Solve for r

$$r=-a\left(\sin(\theta )-999\right)$$