Datrys {l}{sin(6x)=-2/3}{y=(sin(3x)-cos(3x))^2}{text{Solvefor}ztext{where}}{z=y}

$$\left. \begin{array} { l } { \sin(6 x) = -\frac{2}{3} }\\ { y = {(\sin(3 x) - \cos(3 x))} ^ {2} }\\ { \text{Solve for } z \text{ where} } \\ { z = y } \end{array} \right.$$

Datrys ar gyfer x, y, z

$$x=\left(-\frac{1}{6}\right)arcSin(\frac{2}{3})+\frac{1}{3}\pi n_{1}+\frac{1}{3}\pi \text{, }n_{1}\in \mathrm{Z}\text{, }y=1+\frac{5}{16}\left(CosI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)\right)^{3}\left(\left(-1\right)SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)CosI(\frac{1}{6}arcSin(\frac{2}{3}))\right)^{3}+\left(-\frac{3}{32}\right)\left(CosI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)\right)\left(\left(-1\right)SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)CosI(\frac{1}{6}arcSin(\frac{2}{3}))\right)^{5}+\left(-\frac{3}{32}\right)\left(CosI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)\right)^{5}\left(\left(-1\right)SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)CosI(\frac{1}{6}arcSin(\frac{2}{3}))\right)\text{, }n_{1}\in \mathrm{Z}\text{, }z=1+\frac{5}{16}\left(CosI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)\right)^{3}\left(\left(-1\right)SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)CosI(\frac{1}{6}arcSin(\frac{2}{3}))\right)^{3}+\left(-\frac{3}{32}\right)\left(CosI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)\right)\left(\left(-1\right)SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)CosI(\frac{1}{6}arcSin(\frac{2}{3}))\right)^{5}+\left(-\frac{3}{32}\right)\left(CosI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)\right)^{5}\left(\left(-1\right)SinI(\frac{1}{6}arcSin(\frac{2}{3}))\left(CosI(\frac{1}{3}\pi n_{1})+\left(-1\right)SinI(\frac{1}{3}\pi n_{1})\times 3^{\frac{1}{2}}\right)+\left(SinI(\frac{1}{3}\pi n_{1})+3^{\frac{1}{2}}CosI(\frac{1}{3}\pi n_{1})\right)CosI(\frac{1}{6}arcSin(\frac{2}{3}))\right)\text{, }n_{1}\in \mathrm{Z}$$

$$x=\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3})\text{, }n_{2}\in \mathrm{Z}\text{, }y=\frac{1}{16}\left(16+5\left(3^{\frac{1}{2}}\left(CosI(\frac{1}{3}\pi n_{2})CosI(\frac{1}{6}arcSin(\frac{2}{3}))+\left(-1\right)SinI(\frac{1}{3}\pi n_{2})SinI(\frac{1}{6}arcSin(\frac{2}{3}))\right)+\left(-1\right)\left(SinI(\frac{1}{3}\pi n_{2})CosI(\frac{1}{6}arcSin(\frac{2}{3}))+SinI(\frac{1}{6}arcSin(\frac{2}{3}))CosI(\frac{1}{3}\pi n_{2})\right)\right)^{3}\left(CosI(\frac{1}{3}\pi n_{2})CosI(\frac{1}{6}arcSin(\frac{2}{3}))+\left(-1\right)SinI(\frac{1}{3}\pi n_{2})SinI(\frac{1}{6}arcSin(\frac{2}{3}))+\left(SinI(\frac{1}{3}\pi n_{2})CosI(\frac{1}{6}arcSin(\frac{2}{3}))+SinI(\frac{1}{6}arcSin(\frac{2}{3}))CosI(\frac{1}{3}\pi n_{2})\right)\times 3^{\frac{1}{2}}\right)^{3}+\left(-48\right)\left(3^{\frac{1}{2}}CosI(\frac{1}{6}\left(2\pi n_{2}+arcSin(\frac{2}{3})\right))+\left(-1\right)SinI(\frac{1}{6}\left(2\pi n_{2}+arcSin(\frac{2}{3})\right))\right)\left(SinI(\frac{1}{6}\left(\pi +2\pi n_{2}+arcSin(\frac{2}{3})\right))\right)^{5}+\left(-96\right)\left(CosI(\frac{1}{6}\left(\pi +2\pi n_{2}+arcSin(\frac{2}{3})\right))\right)^{5}SinI(\frac{1}{6}\left(\pi +2\pi n_{2}+arcSin(\frac{2}{3})\right))\right)\text{, }n_{2}\in \mathrm{Z}\text{, }z=1+20\left(CosI(\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3}))\right)^{3}\left(SinI(\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3}))\right)^{3}+\left(-6\right)CosI(\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3}))\left(SinI(\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3}))\right)^{5}+\left(-6\right)\left(CosI(\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3}))\right)^{5}SinI(\frac{1}{6}\pi +\frac{1}{3}\pi n_{2}+\frac{1}{6}arcSin(\frac{2}{3}))\text{, }n_{2}\in \mathrm{Z}$$