Understanding the Fourier transform

In a current lecture we had a quick visual introduction to the Fourier transform. It was explained that the Fourier transform turns a given wave into an approximative wave that consists out of cosine waves.

According to the explanation, the graphs below are the FT waves of sine and cosine. To my understanding the FT waves are equivalent to the "frequency spectrums" of the waves. And the frequency spectrum simply gives us the required consine waves with the frequency and its amplitude. The frequency of each wave is denoted on the x-axis and the amplitude on the y-axis.

If I understand it right, shouldn't the waves of sine cancel out since cosine is symmetric?

$$amp *cos(freq \cdot x) + (-amp)*cos(-freq \cdot x) = 0$$ • You'll need to think of your y-axis as being complex to represent phase, but yes, your explanation is much better than what the lecture did. – Marcus Müller May 12 at 20:15
• Fourier Transform is just a correlation: multiply and integrate. Exact phase alignment, and integer multiples of that, achieve maximum correlation. – analogsystemsrf May 12 at 21:09

No, you forget that the Fourier transform is complex. The amplitudes of the two dirac impulses (that's what you call "FT waves"; the term "FT waves" doesn't exist, as far as I know) of the sine are $$\-\frac{1}{2j}\$$ and $$\+\frac{1}{2j}\$$, with $$\j\$$ being the imaginary unit $$\\sqrt{-1}\$$.