Solve F=-mgsinθ

$$F = - m g \sin \theta$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{F}{m\sin(\theta )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\text{ and }m\neq 0\\g\in \mathrm{R}\text{, }&\left(m=0\text{ or }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\right)\text{ and }F=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-m\right)g\sin(\theta )=F$$

The equation is in standard form.

$$\left(-m\sin(\theta )\right)g=F$$

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

$$\frac{\left(-m\sin(\theta )\right)g}{-m\sin(\theta )}=\frac{F}{-m\sin(\theta )}$$

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

$$g=\frac{F}{-m\sin(\theta )}$$

Divide F by -m\sin(\theta ).

$$g=-\frac{F}{m\sin(\theta )}$$

Solve for F

$$F=-gm\sin(\theta )$$