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Trigonometry Cosine Equation

Petrus

Well-known member
Feb 21, 2013
739
Hello MHB,
This is an old exam.


What real x satisfies equation \(\displaystyle 4\cos^2(x)-4=6\cos(x)\)

Progress:
Subsitute \(\displaystyle u=\cos(x)\) and I solve this equation
\(\displaystyle 4u^2-6u-4=0 \)
\(\displaystyle u_1=-\frac{1}{2}\) and \(\displaystyle u_2=2\)
so if we take arccos of them we get
\(\displaystyle x=\frac{3\pi}{2}+2k\pi\) which agree with facit but they got also \(\displaystyle x=-\frac{3\pi}{2}+2k\pi\) which I dont understand also how shall I know what \(\displaystyle x=\cos^{-1}(2)\) is in exam? I am doing something wrong or..?

Regards,
\(\displaystyle |\pi\rangle\)
 

Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
Re: cos equation

Well, $\cos{(x)} = \cos{(x + 2 \pi)}$ by definition ($2 \pi$ is like adding one whole revolution to your angle, so it's the same angle). The inverse cosine function is multivalued, but $\arccos$ is defined as the principal value. Then $\cos{(x)} = \cos{(-x)}$ and so that second value follows (can you see why?)

You need to be careful here because manipulating such multivalued functions can create or destroy solutions to your original equation, so always graph your equation to get a rough idea where the roots are to be sure you didn't miss any and so on.

EDIT: inb4 reply tsunami (Tongueout)
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: cos equation

I think you mean to write one of the solutions is:

\(\displaystyle x=\frac{2\pi}{3}+2k\pi\) where \(\displaystyle k\in\mathbb{Z}\)

and since $\cos(-\theta)=\cos(\theta)$, we also have:

\(\displaystyle x=-\frac{2\pi}{3}+2k\pi\)

which means we may write:

\(\displaystyle x=\frac{2\pi}{3}(6k\pm1)\)
 

Petrus

Well-known member
Feb 21, 2013
739
Re: cos equation

Hello MHB,


Thanks for fast responed and help! I will have to check this more :) I have indeed seen that \(\displaystyle \cos(x)=\cos(-x)\) but I dont think I know why but I will think about this and how does it work with sinus?

Regards,
\(\displaystyle |\pi\rangle\)