How to change duty cycle of $\sin()$ function

Question

Is there a way to change the duty cycle of the sine function?

I am trying to use $\operatorname{sgn}(\sin(x))$ functions and I would love to change the width of pulses.

Thank you in advance!

Answer 1

Originally i thought you wanted to change the frequency of the function:
Think about how the sine function is defined. If i put something in its argument i will get the sine of that thing. Now if i want the function to lengthen the function on the $x$-axis you need to put half of it to get the same function.

See: https://www.desmos.com/calculator/zkx76calgy

For a numerical example: $$\sin\left(a\cdot\pi\right)=0$$

If i say:

$$\sin\left(\frac{b}{2}\pi\right)=0$$

$b$ needs to be twice the value of $a$ to get the same output. Essentially stretching the function sideways.

Manipulating the duty cycle is a digital systems problem and can be done formally easily in that manner. If you want to specifically calculate a specific function like that then the easiest but also the most tedious way to calculate it is to figure out your function between two oscillations and just take the Fourier series of it. But i don't think this is a very useful method for understanding the specific ratio.