Wave motion and a stretched string

[mondo]

I continue my reading of Griffiths electrodynamics (chapter 9, electromagnetic waves) and I got stuck on this:

Author tries to prove a stretched string supports wave motion and I found it very difficult to grasp.
In the second equation, why can we replace sin function with a tangents really? What guarantees that the angles are small? Is he trying to match it to partial derivative formula?
Next, how a difference of partial derivatives $\frac{\partial_{f}}{\partial_{z}}$ became a second partial derivative in the second equation?

Thank you.

 

[pasmith]

It is the string's displacement from its stable equilibrium position (assumed horizontal) that is small (compared with the length of the string); a consequence of this is that the angle between the horizontal and the tangent to the string is everywhere small. If this assumption is not valid, then neither are the results of the analysis.

For your second question, you have from the Taylor series expansion of $\frac{\partial f}{\partial z}$ wrt $z$ that $$\left.\frac{\partial f}{\partial z}\right|_{z+ \Delta z} = \left.\frac{\partial f}{\partial z}\right|_{z} + \left.\frac{\partial^2 f}{\partial z^2}\right|_{z}\Delta z + O(\Delta z^2).$$