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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.)
4 sec2 x + 2 tan2 x − 6 = 0

Respuesta :

mathmate
4sec^2(x)+2tan^2(x)-6=0
using 1+tan^2(x)=sec^2(x)
4(1+tan^2(x))+2tan^2(x)=6
4+4tan^2(x))+2tan^2(x)=6
6tan^2(x)=6-4=2
tan^2(x)=1/3
tan(x)=sqrt(1/3)=1/sqrt(3)=sqrt(3)/3
x=atan(sqrt(3)/3)=pi/6=30 degrees
However, tan(x) is a periodic function with period pi, therefore
the solutions are pi/6, pi/6+pi, or
{pi/6, 7pi/6}
to the risk of sounding redundant

[tex]\bf 1+tan^2(\theta)=sec^2(\theta)\\\\ -------------------------------\\\\ 4sec^2(x)+2tan^2(x)-6=0\implies 4[1+tan^2(x)]+2tan^2(x)=6 \\\\\\ 4+4tan^2(x)+2tan^2(x)=6\implies 6tan^2(x)=2\implies tan^2(x)=\cfrac{1}{3}[/tex]

[tex]\bf tan(x)=\pm\sqrt{\cfrac{1}{3}}\implies tan(x)=\pm\cfrac{1}{\sqrt{3}}\implies tan(x)=\pm\cfrac{\sqrt{3}}{3} \\\\\\ \measuredangle x=tan^{-1}\left(\pm \frac{\sqrt{3}}{3} \right)\implies \measuredangle x= \begin{cases} \frac{\pi }{6}\\\\ \frac{5\pi }{6}\\\\ \frac{7\pi }{6}\\\\ \frac{11\pi }{6} \end{cases}[/tex]

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