Question

15 tan^3 x=5 tan x Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

Answer 1

$\bf 15tan^3(x)=5tan(x)\implies 3tan^3(x)=tan(x) \\\\\\ 3tan^3(x)-tan(x)=0\implies tan(x)[3tan^2(x)-1]=0 \\\\\\ \begin{cases} tan(x)=0\\\\ \measuredangle x=0,\pi \\ ----------\\ 3tan^2(x)-1=0\\ tan^2(x)=\frac{1}{3}\\ tan(x)=\pm\sqrt{\frac{1}{3}}\\\\ tan(x)=\pm\frac{1}{\sqrt{3}}\\\\ \measuredangle x=\frac{\pi }{6}, \frac{5\pi }{6},\frac{7\pi }{6}, \frac{11\pi }{6} \end{cases}$