Fourier Series of Real-valued Functions

Question

Context: For a $2\pi$-periodic bounded function $f:\mathbb{R}\to\mathbb{C}$, we define the complex Fourier coefficients of $f$ by $$ \hat{f_k}:=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}\,dx. $$ We call Fourier series of $f$ the formal series $$ \sum_{k=-\infty}^{\infty}\hat{f_k}e^{ikx}.\tag{1} $$ Now, it is easily shown that $$ \overline{\hat{f_k}}=\hat{\overline{f}}_{-k}. $$ Hence if $f$ is real-valued then $$ \overline{\hat{f_k}}=\hat{f}_{-k}.\tag{2} $$ Reading some notes, it is said that if $f$ is a real-valued function, then we can write $$ \hat{f_k}=\frac{1}{2\pi}\int_0^{2\pi}f(x)\cos(kx)\,dx-i\frac{1}{2\pi}\int_0^{2\pi}f(x)\sin(kx)\,dx.\tag{3} $$ Putting $$ a_k:=\frac{1}{\pi}\int_0^{2\pi}f(x)\cos(kx)\,dx,\\ b_k:=\frac{1}{\pi}\int_0^{2\pi}f(x)\sin(kx)\,dx, $$ where $a_k,b_k$ are called the real Fourier coefficients of $f$, it is said that in view of $(2)$, we have the relations \begin{align} \hat{f_0}&=\frac{1}{2}a_0,\\ \hat{f_k}&=\frac{1}{2}(a_k-ib_k),\\ a_k&=\hat{f_k}+\hat{f}_{-k},\tag{4}\\ b_k&=i(\hat{f_k}-\hat{f}_{-k}),\\ a_{-k}&=a_k,\\ b_{-k}&=-b_k. \end{align} Finally it is said that if, once again, $f$ is real-valued, then we can write $(1)$ as the trigonometric series $$ \frac{1}{2}a_0+\sum_{k=1}^{\infty}a_k\cos(kx)+b_k\sin(kx).\tag{5} $$ Question: Is it really essential that $f$ be real-valued for $(3)$, $(4)$ and $(5)$ to hold? It seems to me that everything holds even if $f$ is complex-valued...