Is there a wave equation which straightens/rounds the lines between troughs and crests?

Question

I'm looking for a periodic wave-shape that can transform from something like a sine wave to a zigzag.

I'm particularly interested in:

• the straightening/rounding of the curve between the crests and troughs or alternatively
• sharpening/rounding the trough/crests

I am not a well studied mathematician, so a brief explanation of how each variable in such an equation would relate to the desired effect.

I found this equation for a zig zag but there is not rounding that occurs y = cos^(-1)(cos(x)) https://www.youtube.com/watch?v=v-C8wsWtdWA

Answer 1

Consider the family of functions $$ f_a(x)=\cos^{-1}(a \cos(x) ) $$ When $a= \pm 1$ it is a zigzag

when $-1<a<1$ it is a wave with rounded crests and troughs.

Here is a link to a Desmos file that graphs the functions using a slider for the value of $a$ ...

Rounding crests