Solve dv/dθ=sinθ+rtanθ

$$\frac { d v } { d \theta } = \sin \theta + r \tan \theta$$

Solve for r

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

Steps for Solving Linear Equation

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

$$\sin(\theta )+r\tan(\theta )=\frac{\mathrm{d}}{\mathrm{d}\theta }(v)$$

Subtract \sin(\theta ) from both sides.

$$r\tan(\theta )=\frac{\mathrm{d}}{\mathrm{d}\theta }(v)-\sin(\theta )$$

The equation is in standard form.

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

Divide both sides by \tan(\theta ).

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

Dividing by \tan(\theta ) undoes the multiplication by \tan(\theta ).

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

Divide -\sin(\theta ) by \tan(\theta ).

$$r=-\cos(\theta )$$