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what values for θ(0<θ<2pi) satisfy this equation? tan^2 θ = 3/2 secθ
 10 months ago
 10 months ago
what values for θ(0<θ<2pi) satisfy this equation? tan^2 θ = 3/2 secθ
 10 months ago
 10 months ago

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satellite73Best ResponseYou've already chosen the best response.1
this is a poser, hold on i think i have an idea
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
it is not that hard, just my first attempt didn't work add \(\frac{3}{2}\sec(x)\) to start withi \[\tan^2(x)+\frac{3}{2}\sec(x)=0\] then rewrite as \[\frac{\sin^2(x)}{\cos^2(x)}+\frac{3}{2\cos(x)}=0\] add up to get \[\frac{2\sin^2(x)+3\cos(x)}{2\cos^2(x)}=0\] then set the numerator equal to zero and solve
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
you get \[2\sin^2(x)+3\cos(x)=0\] rewrite as \[2(1\cos^2(x))+3\cos(x)=0\] and solve the quadratic equation in cosine
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
you good from there or you need more steps?
 10 months ago

infinitemoesBest ResponseYou've already chosen the best response.0
I'm gonna be honest with ya, I have absolutely no idea what I'm doing. So extra steps would be great!
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
ok but before i write them, do you have any questions about the steps i wrote? there was no real trig, just algebra
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
the only trig i used was that \(\tan^2(x)=\frac{\sin^2(x)}{\cos^2(x0}\) and \(\sec(x)=\frac{1}{\cos(x)}\)
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
now we have \[2(1\cos^2(x))+3\cos(x)=0\] which is like solving \[2(1u^2)+3u=0\] rewrite as \[22u^3+3u=0\] or \[2u^23u2=0\] factor as \[(2u+1)(u2)=0\] so \[u=\frac{1}{2}\] or \[u=2\] i.e. \[\cos(x)=\frac{1}{2}\] which is the only solution, because cosine cannot be 2
 10 months ago

satellite73Best ResponseYou've already chosen the best response.1
then look in the unit circle to see for what values of \(x\) you get \[\cos(x)=\frac{1}{2}\] and you will see it is \[x=\frac{2\pi}{3}\] or \[x=\frac{4\pi}{3}\]
 10 months ago

infinitemoesBest ResponseYou've already chosen the best response.0
oh my gosh you are wonderful. thank you so much!
 10 months ago
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