oscillators - Neglecting some wave functions by assuming that the angle between tension force and horizontal is small in the derivation of wave equation in $1D$

In the derivation of the wave equation in classical mechanics in one dimension in a string. It's assumed that the angle between the tension and the horizontal line is small. This is assumed to allow us to let $$\sin (\theta)\approx \theta\approx \tan(\theta)= \frac{\partial y}{\partial x}$$ and complete the derivation.

My Question is, Isn't it true that such an assumption does neglect some wave functions? I mean that those waves in which the angle is not relatively small will not be solutions of the wave equation and so the equation does not consider them. Is this true? If yes, What to do then? If no, could you explain please?