Question

Resolva em R:
tg³(x)-tg(x)=0

me ajudem por favor

Answer 1

$\mathsf{tg^3(x)-tg(x)=0}\\\mathsf{tg(x)(tg^{2}(x)-1)=0}$

$\mathsf{tg(x)=0}\\\mathsf{tg(x)=tg(0)\to~x=k\pi}\\\mathsf{tg(x)=tg(\pi)\to~x=\pi+k\pi}\\\mathsf{tg(x)=tg(2\pi)\to~x=2\pi+k\pi}$

$\mathsf{tg^2(x)-1=0}\\\mathsf{tg^2(x)=1}\\\mathsf{tg(x)=\pm\sqrt{1}}\\\mathsf{tg(x)=\pm1}$

$\mathsf{tg(x)=1}\\\mathsf{tg(x)=tg(\dfrac{\pi}{4})\to~x=\dfrac{\pi}{4}+k\pi}\\\mathsf{tg(x)=tg(\dfrac{5\pi}{4})\to~x=\dfrac{5\pi}{4}+k\pi}$

$\mathsf{tg(x)=-1}\\\mathsf{tg(x)=tg(\dfrac{3\pi}{4})\to~x=\dfrac{3\pi}{4}+k\pi}\\\mathsf{tg(x)=tg(\dfrac{7\pi}{4})\to~x=\dfrac{7\pi}{4}+k\pi}$

A solução da equação é

$s=\{k\pi,\pi+k\pi,2\pi+k\pi,\dfrac{\pi}{4}+k\pi,\dfrac{5\pi}{4}+k\pi,\dfrac{3\pi}{4}+k\pi,\dfrac{7\pi}{4}+k\pi\}$