Solve Y=a(1-sinθ)

$$Y = a ( 1 - \sin \theta )$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{Y}{\sin(\theta )-1}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =2\pi n_{1}+\frac{\pi }{2}\\a\in \mathrm{R}\text{, }&Y=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =2\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a\left(1-\sin(\theta )\right)=Y$$

Use the distributive property to multiply a by 1-\sin(\theta ).

$$a-a\sin(\theta )=Y$$

Combine all terms containing a.

$$\left(1-\sin(\theta )\right)a=Y$$

The equation is in standard form.

$$\left(-\sin(\theta )+1\right)a=Y$$

Divide both sides by 1-\sin(\theta ).

$$\frac{\left(-\sin(\theta )+1\right)a}{-\sin(\theta )+1}=\frac{Y}{-\sin(\theta )+1}$$

Dividing by 1-\sin(\theta ) undoes the multiplication by 1-\sin(\theta ).

$$a=\frac{Y}{-\sin(\theta )+1}$$

Solve for Y

$$Y=a\left(-\sin(\theta )+1\right)$$