I am learning the Fourier Series. It is generally represented as
$$f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx)+\sum_{n=1}^{\infty}\sin(nx).$$
Here Destin represents them as circles ,but to represent Fourier Series as circles we need Complex Exponentials ($e^{ix})$ ,but according to the video he says we use sine terms. Shouldn't we include cosine terms too?
The usage of circle to represent a diagram is beautiful ,we use vectorsthat are rotating ;placed over other vectors that are rotating at different speeds to acheive this ,does $cosine$ $terms$ get included when using circles ?
How is the below mentioned term represented as exponentials ?
This term $f(x)$=$\frac{a_o}{2}$+ $ \sum_{n=1}^{\infty}$ $a_n$cos($nx$)+$\sum_{n=1}^{\infty}$sin($nx$). how can it be represented as an exponential ?
And is there any piece of text or video that explains Fourier Transform in terms of Complex Exponentials in an intuitive manner? I tried understanding Grant's Video on this topic(https://youtu.be/spUNpyF58BY) but it's a bit hard to understand .Is there any other source that explains Fourier Transform intuitively ?