What is a sine wave?

One way would be to describe a sinewave with respect to the unit circle. The radius obviously draws a circle BUT the x and y co-ordinates trace out the familiar waveforms.

This also helps with pictorially explaining Eulers formula:

\$e^{i x} = cos(x)+ i\cdot sin(x)\$

where the special case of \$x = \pi\$ yields Eulers identity: \$e^{i \pi} + 1 = 0\$

(source: https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/)

The easiest explanation I find is encapsulated in the moving image above. It's all about right angle triangles existing inside a circle.

Picture taken from here. See also Why is a sine wave preferred over other waveforms.

Simple: a sine wave in time, t, is the imaginary part of:

$$e^{j \omega t}$$

where ω is the angular frequency.