Solve r=a(t-sint)

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

Solve for a

$$\left\{\begin{matrix}a=\frac{r}{-\sin(t)+t}\text{, }&-\sin(t)+t\neq 0\\a\in \mathrm{R}\text{, }&r=0\text{ and }-\sin(t)+t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a\left(t-\sin(t)\right)=r$$

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

$$at-a\sin(t)=r$$

Combine all terms containing a.

$$\left(t-\sin(t)\right)a=r$$

The equation is in standard form.

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

Divide both sides by t-\sin(t).

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

Dividing by t-\sin(t) undoes the multiplication by t-\sin(t).

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

Solve for r

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