Solve cosθ+sinθ=pir

$$\cos \theta + \sin \theta = \pi r$$

Solve for r (complex solution)

$$r=\frac{\left(1-i\right)e^{i\theta }+\left(1+i\right)e^{-i\theta }}{2\pi }$$

Steps for Solving Linear Equation

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

$$\pi r=\cos(\theta )+\sin(\theta )$$

The equation is in standard form.

$$\pi r=\sin(\theta )+\cos(\theta )$$

Divide both sides by \pi .

$$\frac{\pi r}{\pi }=\frac{\sin(\theta )+\cos(\theta )}{\pi }$$

Dividing by \pi undoes the multiplication by \pi .

$$r=\frac{\sin(\theta )+\cos(\theta )}{\pi }$$

Solve for r

$$r=\frac{\sin(\theta )+\cos(\theta )}{\pi }$$

Steps for Solving Linear Equation

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

$$\pi r=\cos(\theta )+\sin(\theta )$$

The equation is in standard form.

$$\pi r=\sin(\theta )+\cos(\theta )$$

Divide both sides by \pi .

$$\frac{\pi r}{\pi }=\frac{\sin(\theta )+\cos(\theta )}{\pi }$$

Dividing by \pi undoes the multiplication by \pi .

$$r=\frac{\sin(\theta )+\cos(\theta )}{\pi }$$

Solve for θ (complex solution)

$$\theta =-i\ln(\left(\frac{1}{2}+\frac{1}{2}i\right)\left(\sqrt{\left(\pi r\right)^{2}-2}+\pi r\right))+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =-i\ln(\frac{\left(-1-i\right)\sqrt{\left(\pi r\right)^{2}-2}+\pi \left(1+i\right)r}{2})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}$$