I'm sorry to bother you with this, as there may be a very obvious answer, but still: I just stumbled upon some old trigonometry stuff and realized that:

`sin(0π) ≠ 0`

Y

`sin(π/2) ≠ 1`

Y

`sin(3π/2) ≠ -1`

but the answer to all those sinus entries should be `undefined`

Since all these numbers (0, 1, -1) are actually limits, we cannot have a relation of 0 since that would mean that there is an angle 0 that is not possible in the right triangle.

The same applies to 1 and -1 since there cannot be 2 right angles in a triangle :).

then, later:

### the `range`

breast

it is not (-1, 1) but rather: (-1, 0) and (0, 1)

### the `domain`

(sine entries) are:

everybody `x`

in real numbers |

x% π! = 0 # <- otherwise, get a ratio of 0 that is impossible

x% π / 2! = 0 # <- otherwise, get a ratio of 1 that is impossible

x% 2π / 3%! = 0 # <- otherwise, get a ratio of -1 that is impossible

The chart should look like this, with the points excluded from the chart:

sine (x)

My question is: `why my thinking is wrong?`

Otherwise, all those fancy things like Euler's identity won't work for π, for example

e ^ (iπ) = cos (π) + isin (π)

it would mean that:

e ^ (iπ) = undefined + i * undefined

which makes no sense.

`Are all those -1, 0, 1 values just a little helper crutches to keep us going so the math will "somehow" work?`

- the same applies to the cosine function, of course

** sorry for some awkward notations, I'm degenerated by programming