Решете r=2sin(3θ).θ

$$r = 2 \sin ( 3 \theta ) . \theta$$

Решаване за θ (complex solution)

$$\theta =-\frac{i\ln(\frac{i\left(\sqrt{r^{2}-4}+r\right)}{2})}{3}+\frac{2\pi n_{1}}{3}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =-\frac{i\ln(\frac{-i\sqrt{r^{2}-4}+ir}{2})}{3}+\frac{2\pi n_{2}}{3}\text{, }n_{2}\in \mathrm{Z}$$

Решаване за r

$$r=2\sin(3\theta )$$

Решаване за θ

$$\theta =\frac{\arcsin(\frac{r}{2})+2\pi n_{1}}{3}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =\frac{-\arcsin(\frac{r}{2})+2\pi n_{2}+\pi }{3}\text{, }n_{2}\in \mathrm{Z}\text{, }|r|\leq 2$$