## on how to see goniometric functions in space

let be a linear function, like

$$ \tag{1} y=x $$

onto that then another function, that includes the linear function like, increasing the number of dimensions

$$ \tag{2} g(y(x)) = z $$

bc

$$ y=x $$

can be expanded to

$$ y(x) = x $$

so goes a simple function

$$ \tag{3} z = g(y(x)) \in Z(g, y, x)$$

and you starting to form a changing system \(Z\) in, say, time

$$ \tag{4} \frac{d}{dt}z ⇒ z_t’ = \zeta (t) \frac{\partial g(y(x))}{\partial t} \in Z(g, y, x, t, \zeta) $$

$$ \lvert z_t’ \rvert > 0 $$

then if we integrate both sides

$$ \int z_t’ dt = \zeta (t) \int dg(y(x)) $$ $$ z + c_1 = \zeta (t) (z + c_2) $$ $$ z = \frac{\zeta (t) c_2 - c_1}{1 - \zeta (t)} $$

in the end, you have a function/system that is changing in, say, time, or discrete followup

$$ \tag{5} \Delta_k z = g(y(x_{k}) - g(y(x_{k-1})) \in Z(g, y, x, k),~k \in N > 1 $$

$$ \lvert \Delta_k z \rvert > 0 $$

so the linear function in lower dimension

$$y = x$$

could be changing in time somehow (in higher dimensions) with its linearity being the higher function’s object, therefore there should be an other parameter, function that changes in time/angle/idk. \(Q. E. D.\)