Problems with function transforming [duplicate]

So in quantum mechanics, most of the people uses exponential equation to handle the problem and present the relationship. Another word, we often handle the classical wave function in trigonometric way by utilizing sine and cosine. How can these two be correlated?

Using Euler's formula the exponential equation will finally result in an excess imaginary term of $i\sin(kx-\omega t)$. What is the formal process to deal with this?

• As I understand, you are asking about the correlation between the exponential-formalism and the trigonometric-formalism in QM. Actually, they are the same, by the means of the Euler's formula. You don't need to worry about the imaginary term: the coefficients in the expansion of wavefuntions into an orthornomal basis are, in general, complex; and measured physical quantities are in the modulus form (like $\left\langle\psi|H|\psi\right\rangle$ or $\left\langle\psi|\psi\right\rangle$). So, despite the fact that you should work with complex numbers in QM, physical values are always REAL. – Lê Dũng Jul 20 '17 at 16:59

Imagine that you have a state in QM, which can describe any system you wish (e.g. Atom, Oscillator and so on). With the help of QM, the quantity which describes the entire system is called a $\textbf{state vector}$ (if you use Dirac approach, or a wave function if you use the Schrodinger approach) usually denoted with $|\Psi>$ or $\Psi(\textbf{r},t)$ for the latter approach. Your wave-function can have many forms including the exponential that you mentioned, so it can have both REAL and IMAGINARY parts.
The key thing here, as Le Dung already stated is that, the actual PHYSICAL quantity (which has a real meaning, and it is measurable) has always the form $|\Psi|^2$ if you want the probability distribution, or $E=<\Psi|H|\Psi>$ if you want to find the energies, $r=<\Psi|r|\Psi>$ if you want the expectation value of the particle's position(don't take this as it is though, it's just a poor and quick example, you can learn more here) and other relevant quantities. Because of the mathematics behind these operations (which usually just imply a modulus squared of the state vector), the imaginary part would not be "present" anymore, so we end up with a real, physical quantity. More on probability in QM here.