Trigonometric Function

Trigonometric functions are an important type of function in calculus and have periodic features. Understanding the continuity properties of trigonometric functions is essential for studying more complicated types of function.

Basic Properties

A trigonometric function has the following basic properties:

• Domain: $\mathbb{R}$ (for $\sin x$ and $\cos x$)
• Continuity: continuous everywhere on its domain
• Periodicity: has periodic features
• Boundedness: the function values are bounded

Common Trigonometric Functions

1. Sine Function

Properties:

• Domain: $\mathbb{R}$
• Continuous everywhere on $\mathbb{R}$
• Period $2\pi$
• Range: $[-1, 1]$
• The graph is a sine curve

2. Cosine Function

Properties:

• Domain: $\mathbb{R}$
• Continuous everywhere on $\mathbb{R}$
• Period $2\pi$
• Range: $[-1, 1]$
• The graph is a cosine curve

3. Tangent Function

Properties:

• Domain: $x \neq \frac{\pi}{2} + k\pi$ ($k$ an integer)
• Continuous on its domain
• Period $\pi$
• Range: $\mathbb{R}$
• Has a vertical asymptote at every point of discontinuity

4. Cotangent Function

Properties:

• Domain: $x \neq k\pi$ ($k$ an integer)
• Continuous on its domain
• Period $\pi$
• Range: $\mathbb{R}$
• Has a vertical asymptote at every point of discontinuity

Continuity Analysis of Trigonometric Functions

Determining Continuity

1. $\sin x$ and $\cos x$: continuous everywhere on $\mathbb{R}$
2. $\tan x$ and $\cot x$: continuous on their domains, with a vertical asymptote at every point of discontinuity

Proof of Continuity

The continuity of the trigonometric functions rests on the following facts:

1. Power series expansion: $\sin x$ and $\cos x$ can be written as power series
2. Continuity of power series: a power series is continuous within its interval of convergence
3. Continuity of composite functions: the composite of continuous functions is continuous

Graph Characteristics of Trigonometric Functions

Summary of Graph Features

1. Periodicity: trigonometric functions are periodic, so the graph repeats itself
2. Smoothness: on its domain, the graph of a trigonometric function is a smooth curve
3. Boundedness: $\sin x$ and $\cos x$ are bounded, while $\tan x$ and $\cot x$ are unbounded
4. Symmetry: $\sin x$ is an odd function and $\cos x$ is an even function

Continuity of Composite Trigonometric Functions

Basic Theorem

If the function $f(x)$ is continuous at $x_0$, then $\sin(f(x))$ and $\cos(f(x))$ are continuous at $x_0$.

Examples

Example 1: $f(x) = \sin(x^2 + 1)$

Analysis:

• The inner function $h(x) = x^2 + 1$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \sin x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $\mathbb{R}$

Example 2: $f(x) = \cos(\ln x)$

Analysis:

• The inner function $h(x) = \ln x$ is continuous on $(0, +\infty)$
• The outer function $g(x) = \cos x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $(0, +\infty)$

Exercises

Exercise 1

Determine the continuity of $f(x) = \sin(x^2)$ at $x = \pi$.

Exercise 2

Determine the continuity of $f(x) = \cos(x^3)$ at $x = 0$.

Exercise 3

Determine the continuity of $f(x) = \tan(x^2)$ at $x = 0$.

Exercise 4

Determine the continuity of $f(x) = \sin(\frac{1}{x})$ at $x = 0$.

Exercise 5

Determine the continuity of $f(x) = \cos(\ln x)$ at $x = 1$.

Exercise 6

Let $f(x) = \begin{cases} \sin x, & x \leq 0 \\ x + 1, & x > 0 \end{cases}$. Determine the continuity of $f(x)$ at $x = 0$.

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Summary

Symbols Used in This Article