I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an escape-time algorithm. I also see similarities to the Mandelbrot fractal in the results.
However, there's a question that's been bothering me for years. When students learn about sine and cosine, they are typically presented as functions of real numbers that describe the ratios of sides of a right triangle with a given angle as the parameter.
I am well aware that sine and cosine can be easily represented by MacLauren series, as well.
I'm also aware of trigonometry identities that relate sine and cosine to hyperbolic sine and cosine.
My question is, do sine and cosine of complex numbers have anything to do with right-triangles and circles, as sine and cosine of real numbers do?