Properties of $y = \tan x$ and Its Graph
📌 What is the Tangent Function?
The tangent function is one of the six fundamental trigonometric functions. It is defined as the ratio of sine to cosine:
$$\tan x = \frac{\sin x}{\cos x}$$
This definition immediately tells us something crucial: whenever $\cos x = 0$, the tangent function is undefined. This leads to one of the most distinctive features of the tangent graph — its vertical asymptotes.
🔑 Key Properties of $y = \tan x$
| Property | Value |
|---|---|
| Domain | All real numbers except $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer |
| Range | $(-\infty, +\infty)$ — all real numbers |
| Period | $\pi$ (or $180°$) |
| Amplitude | Undefined — the function extends to $\pm\infty$ |
| x-intercepts | $x = n\pi$, where $n$ is any integer (i.e., $0, \pm\pi, \pm 2\pi, \ldots$) |
| y-intercept | $(0, 0)$ |
| Vertical Asymptotes | $x = \frac{\pi}{2} + n\pi$ |
| Symmetry | Odd function: $\tan(-x) = -\tan(x)$ |
| Monotonicity | Strictly increasing on each interval between consecutive asymptotes |
🧠 Understanding the Properties
1. Why is the period $\pi$ instead of $2\pi$?
Unlike $\sin x$ and $\cos x$ which have period $2\pi$, the tangent repeats every $\pi$ radians. This is because:
$$\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \frac{\sin x}{\cos x} = \tan x$$
2. Why are there vertical asymptotes?
At $x = \frac{\pi}{2}$, we have $\cos\left(\frac{\pi}{2}\right) = 0$, so $\tan x = \frac{\sin x}{\cos x}$ involves division by zero. As $x$ approaches $\frac{\pi}{2}$:
- From the left: $\tan x \to +\infty$
- From the right: $\tan x \to -\infty$
3. Odd Function Symmetry
Since $\tan(-x) = -\tan(x)$, the graph has rotational symmetry about the origin (180° rotation). This means the graph in Quadrant III is a rotation of the graph in Quadrant I.