📐 Graphs and Basic Properties of Tangent and Cotangent
1. Building the Graph of $y = \text{tg}\, x$
Step-by-Step Construction
Step 1: Determine the Domain
The function $y = \text{tg}\, x = \dfrac{\sin x}{\cos x}$ is undefined wherever $\cos x = 0$. This happens at:
So the domain excludes $x = \pm\dfrac{\pi}{2},\; \pm\dfrac{3\pi}{2},\; \pm\dfrac{5\pi}{2}, \ldots$
Step 2: Fundamental Period
The tangent function has a fundamental period of $\pi$ (not $2\pi$ like sine and cosine!):
Step 3: Compute Key Values on $\left(-\dfrac{\pi}{2};\; \dfrac{\pi}{2}\right)$
| $x$ | $-\dfrac{\pi}{3}$ | $-\dfrac{\pi}{4}$ | $-\dfrac{\pi}{6}$ | $0$ | $\dfrac{\pi}{6}$ | $\dfrac{\pi}{4}$ | $\dfrac{\pi}{3}$ |
|---|---|---|---|---|---|---|---|
| $\text{tg}\, x$ | $-\sqrt{3} \approx -1.73$ | $-1$ | $-\dfrac{\sqrt{3}}{3} \approx -0.58$ | $0$ | $\dfrac{\sqrt{3}}{3} \approx 0.58$ | $1$ | $\sqrt{3} \approx 1.73$ |
Step 4: Plot Points and Connect
Plot these points on the interval $\left(-\dfrac{\pi}{2};\; \dfrac{\pi}{2}\right)$ and connect them with a smooth curve. Notice that as $x \to \pm\dfrac{\pi}{2}$, the values of $\text{tg}\, x \to \pm\infty$.
Step 5: Shift by $\pi$, $2\pi$, $3\pi$...
Since the period is $\pi$, copy the curve and shift it left and right by $\pi$, $2\pi$, $3\pi$, etc. Each copy sits between two consecutive vertical asymptotes.
Step 6: Draw Vertical Asymptotes
Draw dashed vertical lines at every $x = \dfrac{\pi}{2} + \pi n$ where $n \in \mathbb{Z}$.