MHB Solving wave equation using Fourier Transform

[spideyjj1]

I am having trouble with doing the inverse Fourier transform. Although I can find some solutions online, I don't really understand what was going on, especially the part that inverse Fourier transform of cosine function somehow becomes some dirac delta. I've been stuck on it for 2 hrs...

 

[MathProfessor]

The Fourier transform of f is defined by $F(s)=\int_{-\infty}^{+\infty}f(t)e^{-i2\pi st}dt$. if f(t)=1 let $F_{1}$ be it's Fourier transform for $s\neq0$ you get
$F_{1}(s)=\int_{-\infty}^{+\infty}e^{-i2\pi st}dt=0$ ( odd function ). And for s=0
$F_{1}(0)=\int_{-\infty}^{+\infty}1dt=+\infty$ so
$F_{1}$ is then defined by $F_{1}(s)=0$ if $s\neq0$ and $F_{1}(0)=+\infty$. $F_{1}$ is the Dirac delta "function" ( It's a distribution ). $F_{1}(s)=\delta(s)$
Now if $f(t)= cos(2\pi \omega t)$ then $f(t)=\frac{1}{2}(e^{i2\pi \omega t}+e^{-i2\pi\omega t})$. The Fourier transform of f is then $F(s)= \frac{1}{2}\int_{-\infty}^{+\infty}e^{-i2\pi (s-\omega)t}dt+ \frac{1}{2}\int_{-\infty}^{+\infty}e^{-i2\pi (s+\omega)t}dt= \frac{1}{2}F_{1}(s-\omega)+ \frac{1}{2}F_{1}(s+\omega)= \frac{1}{2}\delta(s-\omega)+ \frac{1}{2}\delta(s+\omega)$