Suppose that we have two functions of a shared parameter $t$:
$$x(t) = \cos(t)+\sin(t)+t$$ $$y(t) = \cos(t)\sin(t)t$$
When I plot $x(t)$ against $y(t)$ I get the sense that the graph might be 'half-of-a-Gaussian' function:
Setting this up algebraically, we have:
$\cos(t)\sin(t)t = a \exp \left\{ -\left( \frac{\cos(t)+\sin(t)+t-b}{c} \right)^2 \right\}$
I have tried tinkering around with parameters in Desmos manually, but no luck. It gets a little tricky that you have to consider 3 parameters at once. I found that a,b,c == 0.46, 2.5, 0.73 was pretty close, but not visually perfect.
Does there exist $a,b,c\in \mathbb{R}$ s.t. $\cos(t)\sin(t)t = a \exp \left\{ -\left( \frac{\cos(t)+\sin(t)+t-b}{c} \right)^2 \right\}$?
@Joe pointed out that this equality cannot hold for all $t$ b/c the LHS alternates in sign whereas the RHS does not. Instead, there appears to be a restricted interval of $x(t)$ where the equality might hold. This tacitly means that there are certain intervals over $t$ for which the equality holds.
If it is still tractable, please consider my question over these certain intervals of $t$.
Looks can be deceiving. You did not plot the function over a long enough interval for $t$! If you plot the function with $t \in [0, 12 \pi]$ you obtain a plot like this:
This is obviously not an exponential function.