Understanding a calculator: what does "sign" mean?

Question

I put the following input into a ti-nspire cx CAS:

and it returned:

What does "sign" mean?

Answer 1

It is the sign of $t$, which is $+1$ if $t$ is positive, $0$ if $t$ is $0$, and $-1$ if $t$ is negative.

Answer 2

$\operatorname{sign}:\mathbb R \to \{-1, 0 , 1\}$.

$$\operatorname{sign}(x) = \begin{cases}-1& \text{if }x < 0\\0&\text{if }x =0\\1&\text{ if } x > 0.\end{cases}$$

Basically, it returns the "sign of $x$".

What this is saying is that if $t > 0$ then we get in the numerator $\pi\cos(\pi t) - \pi|t|\sin(\pi t) = \pi\cos(\pi t) - \pi t\sin(\pi t)$

But if $t < 0$ then we get in the numerator $-\pi\cos(\pi-t) -\pi|t|\sin(\pi t)=-\pi\cos(\pi t) + \pi t\sin(\pi t)$

I guess if I weren't a machine I'd put this as $\pm(\pi\cos(\pi t) - \pi t\sin(\pi t))$ where the $\pm$ is determined by the sign of $t$.

Or simply: $\operatorname{sign}(t)\cdot[\pi\cos(\pi t) - \pi t\sin(\pi t)]$.