# How do I combine two simple curves, to make them either averaged or one limits the amplitude of the other?

This looks like it should be really simple, but unfortunately I'm not quickly understanding the basic lessons I've looked over. I have two simple sine curves here, with periods $p_1$ and $p_2$ (I'm prepared to be wrong on my terminology and notation): $$x = \cos ( x \cdot p_1 ) \qquad x = \sin ( x \cdot p_2 ) \\ p_1 = 94.247 \qquad p_2 = 1.5707\\ 0 \le x \le 1 \quad -1 \le y \le 1$$ I'm trying to combine them such that the resulting curve's amplitude is limited near $0$ at the start and has gradually increasing amplitude until it reaches $1$. The second curve is just an unimportant guiding behavior and seems final, but I'll still be adjusting the period of the first curve $(p_1)$ for more changes. (I don't know how I was supposed to arrive at the period multipliers, but that's not important, the trial-and-error values work fine)

I've been working with curve editors for years, being mindful of the trigonometry behind it. But I haven't had use for the equations in 12 years, I couldn't even remember how to graph and adjust a sine wave. I'm sorry if it looks a bit negligent asking for answers on something basic, but I've spent 12 total hours across 2 days getting this far, on what I was only supposed to spend 2 hours on. I did at least figure out getting values I need by working with $x = \frac{ 2 \pi \arccos ( y )} p$

Have a good day!

Appended sketch: https://i.imgur.com/7bJqpH7.jpg

• It's very hard to understand what you're asking. Can you add a sketch of your desired result? – Rahul Feb 27 '18 at 8:08
• Okay! The resulting curve I'm hoping for is this: i.imgur.com/7bJqpH7.jpg (while still hoping to adjust the smaller frequency, but I still want this 0 to 1 easing) – Tera_GX Feb 27 '18 at 8:54
• Oh, then you can just multiply them: wolframalpha.com/input/… – Rahul Feb 27 '18 at 9:01
• Interesting! Thank you! I did think that would be the case, but I guess I messed something up when I was trying. In fact trying it just now I shuffled solving for x instead of y. Oh my. Thanks for helping! (I don't know etiquette here, if it's proper to still add an Answer post, I'll mark it as good; otherwise thanks for making my day better!) – Tera_GX Feb 27 '18 at 9:10