Solve fcosθ+sqrt{3}sinθ=2r

$$f \cos \theta + \sqrt { 3 } \sin \theta = 2 r$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=-\frac{\sqrt{3}\sin(\theta )-2r}{\cos(\theta )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\\f\in \mathrm{C}\text{, }&r=\frac{\sqrt{3}\sin(\theta )}{2}\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\frac{\pi \left(2n_{1}+1\right)}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \sqrt{3}\sin(\theta ) from both sides.

$$f\cos(\theta )=2r-\sqrt{3}\sin(\theta )$$

The equation is in standard form.

$$\cos(\theta )f=-\sqrt{3}\sin(\theta )+2r$$

Divide both sides by \cos(\theta ).

$$\frac{\cos(\theta )f}{\cos(\theta )}=\frac{-\sqrt{3}\sin(\theta )+2r}{\cos(\theta )}$$

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

$$f=\frac{-\sqrt{3}\sin(\theta )+2r}{\cos(\theta )}$$

Solve for r

$$r=\frac{f\cos(\theta )+\sqrt{3}\sin(\theta )}{2}$$

Steps for Solving Linear Equation

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

$$2r=f\cos(\theta )+\sqrt{3}\sin(\theta )$$

Divide both sides by 2.

$$\frac{2r}{2}=\frac{f\cos(\theta )+\sqrt{3}\sin(\theta )}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$r=\frac{f\cos(\theta )+\sqrt{3}\sin(\theta )}{2}$$