حل r^2cos2θ=1

$$r ^ { 2 } \cos 2 \theta = 1$$

برای r حل کنید (complex solution)

$$r=-\left(\cos(2\theta )\right)^{-\frac{1}{2}}$$

$$r=\left(\cos(2\theta )\right)^{-\frac{1}{2}}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\frac{\pi n_{1}}{2}+\frac{\pi }{4}$$

برای θ حل کنید (complex solution)

$$\theta =\pi n_{1}-\frac{i\ln(\frac{-\sqrt{1-r^{4}}+1}{r^{2}})}{2}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =\pi n_{2}-\frac{i\ln(\frac{\sqrt{1-r^{4}}+1}{r^{2}})}{2}\text{, }n_{2}\in \mathrm{Z}\text{, }r\neq 0$$

برای r حل کنید

$$r=\frac{1}{\sqrt{\cos(2\theta )}}$$

$$r=-\frac{1}{\sqrt{\cos(2\theta )}}\text{, }\exists n_{1}\in \mathrm{Z}\text{ : }\left(\theta >\pi n_{1}+\frac{3\pi }{4}\text{ and }\theta <\pi n_{1}+\frac{5\pi }{4}\right)$$