Solve y=Ycosθ+XsinθRightarrowy=Ycosalpha+X

$$y = Y \cos \theta + X \sin \theta \Rightarrow y = Y \cos \alpha + X$$

Solve for X

$$\left\{\begin{matrix}X=-\frac{Y\cos(\theta )-y}{\sin(\theta )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\\X\in \mathrm{R}\text{, }&y=Y\cos(\theta )\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Y\cos(\theta )+X\sin(\theta )=y$$

Subtract Y\cos(\theta ) from both sides.

$$X\sin(\theta )=y-Y\cos(\theta )$$

The equation is in standard form.

$$\sin(\theta )X=-Y\cos(\theta )+y$$

Divide both sides by \sin(\theta ).

$$\frac{\sin(\theta )X}{\sin(\theta )}=\frac{-Y\cos(\theta )+y}{\sin(\theta )}$$

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

$$X=\frac{-Y\cos(\theta )+y}{\sin(\theta )}$$

Solve for Y

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

Steps for Solving Linear Equation

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

$$Y\cos(\theta )+X\sin(\theta )=y$$

Subtract X\sin(\theta ) from both sides.

$$Y\cos(\theta )=y-X\sin(\theta )$$

The equation is in standard form.

$$\cos(\theta )Y=-X\sin(\theta )+y$$

Divide both sides by \cos(\theta ).

$$\frac{\cos(\theta )Y}{\cos(\theta )}=\frac{-X\sin(\theta )+y}{\cos(\theta )}$$

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

$$Y=\frac{-X\sin(\theta )+y}{\cos(\theta )}$$