Solve sin(-x)=ksinx+1

$$\sin ( - x ) = k \sin x + 1$$

Solve for k (complex solution)

$$k=\frac{-\sin(x)-1}{\sin(x)}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}$$

Steps for Solving Linear Equation

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

$$k\sin(x)+1=\sin(-x)$$

Subtract 1 from both sides.

$$k\sin(x)=\sin(-x)-1$$

The equation is in standard form.

$$\sin(x)k=-\sin(x)-1$$

Divide both sides by \sin(x).

$$\frac{\sin(x)k}{\sin(x)}=-\frac{\sin(x)+1}{\sin(x)}$$

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

$$k=-\frac{\sin(x)+1}{\sin(x)}$$

Divide -\left(\sin(x)+1\right) by \sin(x).

$$k=-\left(\frac{1}{\sin(x)}+1\right)$$

Solve for k

$$k=-\frac{1}{\sin(x)}-1$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}$$

Steps for Solving Linear Equation

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

$$k\sin(x)+1=\sin(-x)$$

Subtract 1 from both sides.

$$k\sin(x)=\sin(-x)-1$$

The equation is in standard form.

$$\sin(x)k=-\sin(x)-1$$

Divide both sides by \sin(x).

$$\frac{\sin(x)k}{\sin(x)}=-\frac{\sin(x)+1}{\sin(x)}$$

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

$$k=-\frac{\sin(x)+1}{\sin(x)}$$

Divide -\left(\sin(x)+1\right) by \sin(x).

$$k=-\left(\frac{1}{\sin(x)}+1\right)$$

Solve for x

$$x=\arcsin(\frac{1}{k+1})+2\pi n_{1}+\pi \text{, }n_{1}\in \mathrm{Z}$$

$$x=-\arcsin(\frac{1}{k+1})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }k\geq 0\text{ or }k\leq -2$$