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If you want to take DFT of a cosine wave $cos(\theta)$ sampled at N equally spaced values between $[0, 2\pi]$, then you need to consider taking N-point DFT of the sequence : $$x[n] = cos[2\pi \frac{n}{N}], n = 0,1,2,...,N-1$$ $cos(\theta)$"> And, for this $x[n]$, you dont even have to apply the DFT formula. It can be done pretty simply by using the Euler's ...
Myth: DTFT is Sinc-interpolated DFT. Problem with the above statement: Sinc is not $2\pi$-Periodic function, but all DTFTs are. Correct Answer: Theoretical, Continuous-$\omega$ $2\pi$-Periodic DTFT can be obtained by continuous Lagrangian-interpolation of the DFT Samples. So that the values at $\omega = 2\pi k/N$ will be the DFT Samples $X[k]$ for \$k=0,1,...