I have a query regarding the motivation of complex numbers.
If we expand the functions $e^{ix},\cos x$ & $\sin x$ in terms of Taylor series, then comparing real and imaginary parts we get $e^{ix}=\cos x+i\sin x.$ With the help of this we can express $\cos x$ and $\sin x$ in terms of $e^{ix}$ and $e^{-ix}.$ But what is the geometrical interpretation of this expression? i.e. Why the exponential function can be expressed in terms of trigonometric functions?
In particular, any equation of an oscillation contains either sine function or cosine function or both. Mathematically due to Hooks law the restoring force is proportional to negative displacement(since displacement tends to the equilibrium position). The solution to the corresponding differential equation involves the imaginary number `i' and the trigonometric functions. But what is the philosophical explanation behind this? Without this mathematical computation how can we explain the involvement of imaginary axis in oscillatory motion geometrically? Will the oscillatory curve be considered as two-dimensional only? Since trigonometric functions are involved, what is the significance of the corresponding angle?
Please help me to understand the concept properly. Thanks in advance.