Find solutions with \(x\in [0,2\pi]\) to the equation \(5\sec(x)=3\tan^2(x)+1 . \)

Okay I had an idea for this question:

First, we turn the 3 sec(x) into 3/cos(x). After that, turn 3tan^2(x) into (3 * 1-cos(x))/ 1 + cos(x). I don't know where to go from here. Help needed :D

MobiusLoops Oct 23, 2021

#1**+1 **

$\sec(x) = \frac{1}{\cos(x)}$

\(\begin{align*} \tan^2(x) &= \left(\frac{\sin x}{\cos x} \right)^2 \\ &= \frac{\sin^2 x}{\cos^2 x} \\ &= \frac{\sin^2 x}{\cos^2 x} + \frac{\cos^2 x}{\cos^2 x} - 1\\ &= \frac{\sin^2 x + \cos^2 x}{\cos^2 x} - 1\\ &= \frac{1}{\cos^2 x} - 1 \end{align*}\)

$\frac{5}{\cos(x)} = \frac{3}{\cos^2 x} - 2$

$5 \cos(x) = 3 - 2 \cos^2 x$

$2 \cos^2 x + 5 \cos x - 3 = 0$

$(\cos x + 3) (2 \cos x - 1) = 0$

$\cos x = -3$

$\cos x = \frac{1}{2}$

I think you can take it from here.

MathProblemSolver101 Oct 23, 2021

#2**+1 **

I'm still a bit confused. So I would then know that one of the answers in pi/3. But I don't know what other answers there are...

MobiusLoops
Oct 23, 2021

#3**+1 **

Hi Mobius,

You are correct. There is a second answer to the question, but does it fit in the range?

Best,

MPS

MathProblemSolver101
Oct 23, 2021