Solve 2xcos(θ)+3xsin(θ)=5

$$2x \cos ( \theta ) +3x \sin ( \theta ) =5$$

Solve for x (complex solution)

$$x=\frac{10}{\left(2-3i\right)e^{i\theta }+\left(2+3i\right)e^{-i\theta }}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\pi -\arctan(\frac{2}{3})$$

Steps for Solving Linear Equation

Combine all terms containing x.

$$\left(2\cos(\theta )+3\sin(\theta )\right)x=5$$

Divide both sides by 2\cos(\theta )+3\sin(\theta ).

$$\frac{\left(2\cos(\theta )+3\sin(\theta )\right)x}{2\cos(\theta )+3\sin(\theta )}=\frac{5}{2\cos(\theta )+3\sin(\theta )}$$

Dividing by 2\cos(\theta )+3\sin(\theta ) undoes the multiplication by 2\cos(\theta )+3\sin(\theta ).

$$x=\frac{5}{2\cos(\theta )+3\sin(\theta )}$$

Solve for x

$$x=\frac{5}{2\cos(\theta )+3\sin(\theta )}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}-\arcsin(\frac{2\sqrt{13}}{13})+\pi$$

Steps for Solving Linear Equation

Combine all terms containing x.

$$\left(2\cos(\theta )+3\sin(\theta )\right)x=5$$

Divide both sides by 2\cos(\theta )+3\sin(\theta ).

$$\frac{\left(2\cos(\theta )+3\sin(\theta )\right)x}{2\cos(\theta )+3\sin(\theta )}=\frac{5}{2\cos(\theta )+3\sin(\theta )}$$

Dividing by 2\cos(\theta )+3\sin(\theta ) undoes the multiplication by 2\cos(\theta )+3\sin(\theta ).

$$x=\frac{5}{2\cos(\theta )+3\sin(\theta )}$$

Solve for θ (complex solution)

$$\theta =2\pi n_{1}-i\ln(\frac{\left(-2-3i\right)\sqrt{25-13x^{2}}+\left(10+15i\right)}{x})+\ln(13)i\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =2\pi n_{2}-i\ln(\frac{\left(2+3i\right)\sqrt{25-13x^{2}}+\left(10+15i\right)}{13x})\text{, }n_{2}\in \mathrm{Z}\text{, }x\neq 0$$