1. ## [SOLVED] Trig equation

Hello

Im having problems to solve the following trig equation.

$\displaystyle cosx-\frac{\frac{1}{2}}{cosx}=\frac{1}{2}$

Im given the following information

$\displaystyle x_{1}=2n\pi$

$\displaystyle x_{2}=\frac{a\pi }{b}+2n\pi$

$\displaystyle x_{3}=\frac{c\pi }{d}+2n\pi$

0 < $\displaystyle \frac{a\pi }{b}+2n\pi$ < $\displaystyle \frac{c\pi }{d}+2n\pi$ < $\displaystyle 2\pi$

where n are integers

im not allowed to use a calculator, so im unsure how to eaven start...

2. What we have to solve or find here?

3. I have to find a, b, c and d

$\displaystyle x_{2}=\frac{a\pi }{b}+2n\pi$

$\displaystyle x_{3}=\frac{c\pi }{d}+2n\pi$

4. Originally Posted by JannetheSwede

Im having problems to solve the following trig equation.

cos x-(1/2)/cos x=1/2

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

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

$\displaystyle 2\cos{x} - 1 = \cos{x}$

$\displaystyle \cos{x} = 1$

$\displaystyle x = 2k\pi$ , $\displaystyle k \in \mathbb{Z}$

5. Originally Posted by skeeter
$\displaystyle \frac{\cos{x} - \frac{1}{2}}{\cos{x}} = \frac{1}{2}$

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

$\displaystyle 2\cos{x} - 1 = \cos{x}$

$\displaystyle \cos{x} = 1$

$\displaystyle x = 2k\pi$ , $\displaystyle k \in \mathbb{Z}$
Yea thats one of three awnsers, but what i'm looking for are the other two... :S

btw, just to clearify $\displaystyle cosx-\frac{\frac{1}{2}}{cosx}=\frac{1}{2}$

6. Originally Posted by JannetheSwede
Yea thats one of three awnsers, but what i'm looking for are the other two... :S
what makes you think there are more solutions?

7. ok, i have changed the initial question to clearify what it was

8. Originally Posted by JannetheSwede
ok, i have changed the initial question to clearify what it was
$\displaystyle \cos{x} - \frac{\frac{1}{2}}{\cos{x}} = \frac{1}{2}$

multiply every term by $\displaystyle 2\cos{x}$ ...

$\displaystyle 2\cos^2{x} - 1 = \cos{x}$

$\displaystyle 2\cos^2{x} - \cos{x} - 1 = 0$

$\displaystyle (2\cos{x} + 1)(\cos{x} - 1) = 0$

$\displaystyle \cos{x} = -\frac{1}{2}$ ... $\displaystyle \cos{x} = 1$

you finish up.

9. $\displaystyle 2cos^2x-cosx-1=0$

$\displaystyle cosx=1,\frac{-1}{2}$

Spoiler:

$\displaystyle cosx=1$ gives $\displaystyle x=2n\pi$

$\displaystyle cosx=\frac{-1}{2}$ gives $\displaystyle x=\frac{2\pi}{3},\frac{4\pi}{3}$

10. Ummh, or no... I still don't get it

I tried to apply this on another problem of the same type

$\displaystyle cosx+\frac{\frac{1}{2}}{cosx}=\frac{3}{2}$

but I was unable to solve it...

I did like this

$\displaystyle 2cos^{2}x+cos x=3cos x$
$\displaystyle 2cos^{2}x-2cos x=0$

and thats how far I got... what do I do from there?

11. Originally Posted by JannetheSwede
Ummh, or no... I still don't get it

I tried to apply this on another problem of the same type

$\displaystyle cosx+\frac{\frac{1}{2}}{cosx}=\frac{3}{2}$

but I was unable to solve it...

I did like this

$\displaystyle 2cos^{2}x+cos x=3cos x$
$\displaystyle 2cos^{2}x-2cos x=0$

and thats how far I got... what do I do from there?

you are wrong .

It simplifies to

$\displaystyle 2cos^2x-3cosx+1=0$

$\displaystyle (2cosx-1)(cosx-1)=0$

$\displaystyle cosx=\frac{1}{2}$ , $\displaystyle cosx=1$