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### IB Maths SL Trigonometric equations

Posted: Sat Apr 13, 2013 4:43 am
IB Mathematics SL– Trigonometry, Trigonometric equations

How can we solve the following trigonometric equation?

$2cos^2x=cosx+1$ for $- \pi \leq x \leq \pi$

Thanks

### Re: IB Maths SL Trigonometric equations

Posted: Sat Apr 13, 2013 4:51 am
IB Maths SL – Trigonometry, Trigonometric equations

By treating the equation as a quadratic in $cosx$ and then factoring.

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

$\Leftrightarrow (2cosx+1)(cosx-1)=0 \Leftrightarrow cosx-1=0 \ or \ 2cosx+1 =0 \Leftrightarrow$

Now, you have to solve two trigonometric equations

$cosx-1=0$ and $2cosx+1 =0$ in the interval $[-\pi , \pi]$

For the first equation we have:

$cosx-1=0\Leftrightarrow cosx=1$

The above equation has only one solution in the interval $[-\pi , \pi]$

$x=0$

For the second equation we have:

$2cosx+1 =0 \Leftrightarrow cosx=-\frac{1}{2}$

The above equation has two solutions in the interval $[-\pi , \pi]$

$x=-\pi +\frac{\pi}{3}=-\frac{2\pi}{3} \ or\ x=\pi-\frac{\pi}{3}=\frac{2\pi}{3}$

Hope these help!!

### Re: IB Maths SL Trigonometric equations

Posted: Sat Apr 13, 2013 5:07 am
Thanks a lot!