A trigonometric polynomial can be written as: $$\frac{a_0}{2} + \sum_{k=1}^{n}(a_k \cos(kx) + b_k \sin(kx))$$
My math books says, that ANY trigonometric polynomial can be written in euler's form: $$\sum_{k=1}^{n}a_k e^{ikx} = \frac{a_0}{2} + \sum_{k=1}^{n}(a_k \cos(kx) + i b_k \sin(kx))$$
Using this form, however, the coefficients must be complex.
My question is now, given the fact that $a_k $ must be complex and sin is imaginary using the euler's form, is it really possible to rewrite ANY (real-valued) trigonometric polynomial from the cos-sin form in the euler form for simplicity? If yes, how is it possible? In the second definition the coefficients are complex and sin is imaginary.