y = A tan(ωx + φ)

Because $\tan x = \sin x/\cos x$ and $\cos x = 0$ at $x = \pi/2, 3\pi/2, \ldots$. At those points the denominator vanishes, so $\tan x$ shoots to $\pm\infty$. Between asymptotes the curve rises monotonically from $-\infty$ to $+\infty$ over a period of $\pi$.

Slide the scaling factor and watch the asymptotes stay locked at odd multiples of $\pi/2$. The lab also reports $\tan(\pi/4) = 1$ and $\tan(\pi/3) = \sqrt{3}$ as sanity checks. Notice how zooming near an asymptote shows the slope blowing up.

Because $\sin(x+\pi) = -\sin x$ and $\cos(x+\pi) = -\cos x$, the two negatives cancel in the ratio: $\tan(x+\pi) = \tan x$. Students writing $2\pi$ on board exams lose marks. The lab compresses two cycles in $0$ to $2\pi$ so this is visually obvious.

Tangent measures slope or steepness. If a slope rises by $h$ over a horizontal distance $d$, the angle of elevation is $\theta = \arctan(h/d)$. Surveyors at construction sites, GPS-based gradient calculators and projectile range formulas all rest on $\tan\theta$.