Graphs of Tangent Function

Overview

The graph of the tangent function $\tan(x)$ is a periodic function with unique characteristics, such as vertical asymptotes and a period of $\pi$. This note outlines its properties and applications in trigonometry.

---

Key Properties of the Tangent Function

Periodicity

The tangent function repeats every $\pi$

$$\tan(x + \pi) = \tan(x)$$

Domain

The function is undefined where $\cos(x) = 0$

$$x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$$

Range

The range of $\tan(x)$ is all real numbers:

$$\text{Range: } (-\infty, \infty)$$

Vertical Asymptotes

Vertical asymptotes occur at

$$x = \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z}$$

Key Points

• $\tan(0) = 0$
• Symmetric about the origin: $\tan(-x) = -\tan(x)$

---

Graph Characteristics

• The graph rises from $-\infty$ to $\infty$ between consecutive vertical asymptotes.
• One period of $\tan(x)$ extends from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$
• The graph has no maximum or minimum values because it is unbounded.

---

Worked Examples

---

---

Summary

• Period: $\pi$
• Domain: Excludes $x = \frac{\pi}{2} + n\pi$
• Range: All real numbers $(-\infty, \infty)$
• Vertical Asymptotes: At $x = \frac{\pi}{2} + n\pi$
• The tangent function graph is periodic and unbounded. Understanding the graph of $\tan(x)$ is crucial for solving trigonometric equations and analyzing periodic phenomena.