Fourier transformation - connection between exponential and trigonomeric forms

Question

On Wikipedia i have come across a Fourier transformation equation in exponential form and its inverse (Wiki):

$$ \begin{split} \mathcal{F}(x) &= \int\limits^{\infty}_{-\infty}\mathcal{f}(k) \, e^{2 \pi i kx} \, \textrm{d} k\\ \mathcal{f}(k) &= \int\limits^{\infty}_{-\infty}\mathcal{F}(x) \, e^{2 \pi i kx} \, \textrm{d} x \end{split} $$

but i allso found that there is a trigonometric form of Fourier transformation (PDF, page 2)

$$ \begin{split} \mathcal{F}(x) &= \int\limits^{\infty}_{-\infty} f(k) \cos(kx) \, \textrm{d}k\\ \mathcal{f}(k) &= \int\limits^{\infty}_{-\infty} \mathcal{F}(x) \cos(kx) \, \textrm{d}x \end{split} $$

MAIN QUESTION:

Could someone show me, how these pairs of equations are connected?

SUB QUESTION:

(i) I think that $\textrm{d}x$ is used for spatial integration (please correct me if i am wrong).

(ii) I think that $\textrm{d}k$ is used for integration over wave vector (please correct me if i am wrong) .

Answer 1

Do you know about Euler's Formula?

$e^{ix} = cos(x) + isin(x)$

I think that's a hint...

Also, see this formula http://en.wikipedia.org/wiki/Sine_and_cosine_transforms#Fourier_inversion The first one in this column is simply the fourier integral applied over the above relation. Everything else is just a simplification into cosine

And this is even better...a whole justification of the connection http://en.wikipedia.org/wiki/Sine_and_cosine_transforms#Relation_with_complex_exponentials