Blog: Find all solutions to the equation cos(3z+π)=0.

Friday, June 26, 2015

Hello!

As you probably know, there are many solutions of the equation $\displaystyle{\cos{{\left({w}\right)}}}={0}.$

The general solution is $\displaystyle{w}=\pm\frac{\pi}{{2}}+{2}{k}\pi,$

where $\displaystyle{k}$ is any integer. Without $\displaystyle\pm$ it may be written as two sequences,

$\displaystyle{w}_{{1}}=\frac{\pi}{{2}}+{2}{k}\pi$ and $\displaystyle{w}_{{2}}=-\frac{\pi}{{2}}+{2}{k}\pi.$

In our problem $\displaystyle{w}={3}{z}+\pi,$ so

$\displaystyle{3}{z}+\pi=\frac{\pi}{{2}}+{2}{k}\pi$ or $\displaystyle{3}{z}+\pi=-\frac{\pi}{{2}}+{2}{k}\pi.$

These equations are linear for $\displaystyle{z}$ and may be solved easily:

$\displaystyle{z}_{{1}}=-\frac{\pi}{{6}}+\frac{{{2}{k}\pi}}{{3}}$ and $\displaystyle{z}_{{2}}=-\frac{\pi}{{2}}+\frac{{{2}{k}\pi}}{{3}}.$

This is the answer (remember that $\displaystyle{k}$ is...