Solve f(x)=x-1/3sin2x+operatorname{asin}x

$$f ( x ) = x - \frac { 1 } { 3 } \sin 2 x + \operatorname { asin } x$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sin(2x)+3fx-3x}{3\sin(x)}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\\a\in \mathrm{R}\text{, }&x=0\text{ or }\left(f=\frac{-\frac{\sin(2x)}{x}+3}{3}\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x-\frac{1}{3}\sin(2x)+a\sin(x)=fx$$

Subtract x-\frac{1}{3}\sin(2x) from both sides.

$$a\sin(x)=fx-\left(x-\frac{1}{3}\sin(2x)\right)$$

To find the opposite of x-\frac{1}{3}\sin(2x), find the opposite of each term.

$$a\sin(x)=fx-x+\frac{1}{3}\sin(2x)$$

The equation is in standard form.

$$\sin(x)a=\frac{\sin(2x)}{3}+fx-x$$

Divide both sides by \sin(x).

$$\frac{\sin(x)a}{\sin(x)}=\frac{\sin(2x)+3x\left(f-1\right)}{3\sin(x)}$$

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

$$a=\frac{\sin(2x)+3x\left(f-1\right)}{3\sin(x)}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{\frac{3a\sin(x)}{x}-\frac{\sin(2x)}{x}+3}{3}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply -1 and \frac{1}{3} to get -\frac{1}{3}.

$$fx=x-\frac{1}{3}\sin(2x)+a\sin(x)$$

The equation is in standard form.

$$xf=a\sin(x)-\frac{\sin(2x)}{3}+x$$

Divide both sides by x.

$$\frac{xf}{x}=\frac{\sin(x)\left(-2\cos(x)+3a\right)+3x}{3x}$$

Dividing by x undoes the multiplication by x.

$$f=\frac{\sin(x)\left(-2\cos(x)+3a\right)+3x}{3x}$$