# Solve the equation of Cos3(theta) = 1/2

• Jun 10th 2012, 10:38 AM
DeeRose
Solve the equation of Cos3(theta) = 1/2
Solve the equation of Cos3(theta) = 1/2 , for theta is an element of real numbers, where (theta) is in radius.

I feel like it's simple but I don't know how :-/
• Jun 10th 2012, 10:51 AM
Goku
Re: Solve the equation of Cos3(theta) = 1/2
$\displaystyle cos(3\theta) = \frac{1}{2}$

$\displaystyle 3\theta = cos^{-1}(\frac{1}{2})$

$\displaystyle 3\theta =60$ degrees

$\displaystyle \theta = 20$ degrees $\displaystyle + 360k$ where k is any integer.
• Jun 10th 2012, 10:52 AM
Reckoner
Re: Solve the equation of Cos3(theta) = 1/2
Quote:

Originally Posted by DeeRose
Solve the equation of Cos3(theta) = 1/2 , for theta is an element of real numbers, where (theta) is in radius.

Is this supposed to be $\displaystyle \cos3\theta=\frac12$ or $\displaystyle \cos^3\theta = \frac12\mathrm?$ If the latter, are you looking for an exact solution?
• Jun 10th 2012, 10:53 AM
Reckoner
Re: Solve the equation of Cos3(theta) = 1/2
Quote:

Originally Posted by Goku
$\displaystyle \theta = 20$ degrees

Your procedure is good, but this is only one of infinitely many solutions.
• Jun 10th 2012, 11:20 AM
DeeRose
Re: Solve the equation of Cos3(theta) = 1/2
It's Cos3(theta) ..Not Cos^3(theta) Is Goku's response the right one then?
• Jun 10th 2012, 11:30 AM
Reckoner
Re: Solve the equation of Cos3(theta) = 1/2
Quote:

Originally Posted by DeeRose
It's Cos3(theta) .. Is Goku's response the right one then?

Note that cosine is equal to $\displaystyle \frac12$ for any angle that is coterminal with $\displaystyle \frac\pi3$ or $\displaystyle -\frac\pi3$. Thus

$\displaystyle \cos3\theta=\frac12$

$\displaystyle \Rightarrow3\theta = 2k\pi\pm\frac\pi3$

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

$\displaystyle \Rightarrow\theta = \frac{(6k\pm1)\pi}9$

where $\displaystyle k\in\mathbb{Z}$.
• Jun 10th 2012, 12:22 PM
DeeRose
Re: Solve the equation of Cos3(theta) = 1/2
Thanks alot! :-)