A way to write trig functions normally without sin cos and tan

You can derive them from Euler’s equation: sin(x) = (e^(ix) - e^(-ix))/(2i) and cos(x) = (e^(ix) + e^(-ix))/2. It’s how you can prove sin(ix) = isinh(x) and cos(ix) = cosh(x).
Some people define the trigonometric functions with the Taylor Series though
You can write it out as exponential functions, but that also came after.

The trig functions are defined, not by equations, but by the circle.

If you travel a certain angle around the circle (starting at the right most edge), then sin is your vertical position and cos is your horizontal position.

We write them as sin and cos, because like pi, it is not easy to express the horizontal and vertical distances as you travel around a circle, but you'd best believe it comes up a lot in the real world.