The Fourier Transform of the function $\cos(n \omega t)$ can be derived by expressing the cosine function in terms of complex exponentials.

1. Expression in Complex Exponentials

We start with: $\cos(n \omega t) = \frac{e^{i n \omega t} + e^{-i n \omega t}}{2}$

2. Fourier Transform of $\cos(n \omega t)$

The Fourier Transform of a function $f(t)$ is defined as: $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt$

Substitute $f(t) = \cos(n \omega t) = \frac{e^{i n \omega t} + e^{-i n \omega t}}{2}$: $F(\omega) = \int_{-\infty}^{\infty} \frac{e^{i n \omega t} + e^{-i n \omega t}}{2} e^{-i \omega t} \, dt$

This expands as: $F(\omega) = \frac{1}{2} \int_{-\infty}^{\infty} e^{i (n \omega - \omega) t} \, dt + \frac{1}{2} \int_{-\infty}^{\infty} e^{-i (n \omega + \omega) t} \, dt$

3. Solving the Integrals

These integrals are known as Dirac delta functions in the frequency domain, resulting in: $F(\omega) = \pi \left[ \delta(\omega - n \omega) + \delta(\omega + n \omega) \right]$

Final Result

Thus, the Fourier Transform of $\cos(n \omega t)$ is: $F(\omega) = \pi \left[ \delta(\omega - n \omega) + \delta(\omega + n \omega) \right]$

This result shows that the Fourier Transform of $\cos(n \omega t)$ has two spikes at frequencies $\omega = \pm n \omega$.

Would you like further details on this derivation or applications of this result?

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Here are five related questions:

1. How would the Fourier Transform change if we used $\sin(n \omega t)$ instead of $\cos(n \omega t)$?
2. What is the Fourier Transform of $\cos(\omega t)$ without the factor $n$?
3. How can this result be used in analyzing periodic signals in signal processing?
4. What are Dirac delta functions, and how are they used in Fourier analysis?
5. How does the Fourier Transform of a cosine compare to that of an exponential function?

Tip: The Fourier Transform of cosines and sines gives a good understanding of frequency components, crucial in analyzing real-world signals like sound waves and electrical currents.