operatornamean^3θ-3tanθ=0 lösen

$$\operatorname { an } ^ { 3 } \theta - 3 \tan \theta = 0$$

Nach a auflösen (komplexe Lösung)

$$\left\{\begin{matrix}a=-\frac{\sqrt[3]{3}\theta ^{-\frac{1}{3}}\sqrt[3]{-\tan(\theta )}}{n}\text{; }a=\frac{\sqrt[3]{3}e^{\frac{\pi i}{3}}\theta ^{-\frac{1}{3}}\sqrt[3]{-\tan(\theta )}}{n}\text{; }a=-\frac{\sqrt[3]{3}ie^{\frac{\pi i}{6}}\theta ^{-\frac{1}{3}}\sqrt[3]{-\tan(\theta )}}{n}\text{, }&n\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\theta \neq 0\\a\in \mathrm{C}\text{, }&\left(\theta =0\text{ or }n=0\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\theta =\pi n_{2}\end{matrix}\right,$$

Nach n auflösen (komplexe Lösung)

$$\left\{\begin{matrix}n=-\frac{\sqrt[3]{3}\theta ^{-\frac{1}{3}}\sqrt[3]{-\tan(\theta )}}{a}\text{; }n=\frac{\sqrt[3]{3}e^{\frac{\pi i}{3}}\theta ^{-\frac{1}{3}}\sqrt[3]{-\tan(\theta )}}{a}\text{; }n=-\frac{\sqrt[3]{3}ie^{\frac{\pi i}{6}}\theta ^{-\frac{1}{3}}\sqrt[3]{-\tan(\theta )}}{a}\text{, }&a\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\theta \neq 0\\n\in \mathrm{C}\text{, }&\left(\theta =0\text{ or }a=0\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\theta =\pi n_{2}\end{matrix}\right,$$

Nach a auflösen

$$\left\{\begin{matrix}a=\frac{\sqrt[3]{\frac{3\tan(\theta )}{\theta }}}{n}\text{, }&n\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\theta \neq 0\\a\in \mathrm{R}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\left(n=0\text{ or }\theta =0\right)\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\theta =\pi n_{2}\text{, }not(n_{2}=0)\end{matrix}\right,$$

Nach n auflösen

$$\left\{\begin{matrix}n=\frac{\sqrt[3]{\frac{3\tan(\theta )}{\theta }}}{a}\text{, }&a\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\theta \neq 0\\n\in \mathrm{R}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\text{ and }\left(a=0\text{ or }\theta =0\right)\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\theta =\pi n_{2}\text{, }not(n_{2}=0)\end{matrix}\right,$$