Continuity of Trigonometric Functions

Trigonometric functions are an important type of function in calculus; understanding their continuity properties is essential for studying more complicated types of function.

Basic Properties

Trigonometric functions have the following basic properties:

• Periodicity: have periodic features
• Continuity: $\sin x$ and $\cos x$ are continuous on $\mathbb{R}$
• Discontinuity: $\tan x$ and $\cot x$ are discontinuous at certain points
• Graph features: a smooth curve, with no jumps or breaks

Continuous Trigonometric Functions

The Sine and Cosine Functions

Sine function: $f(x) = \sin x$

Properties:

• Domain: $\mathbb{R}$
• Continuous on $\mathbb{R}$
• Period $2\pi$
• Range: $[-1, 1]$

Cosine function: $f(x) = \cos x$

Properties:

• Domain: $\mathbb{R}$
• Continuous on $\mathbb{R}$
• Period $2\pi$
• Range: $[-1, 1]$

Proof of Continuity

Proof Strategy:

1. Use the definitions of the trigonometric functions and the properties of limits
2. Prove continuity at an arbitrary point $x_0$
3. Use the periodicity of the trigonometric functions

Discontinuous Trigonometric Functions

The 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

Analysis of the points of discontinuity:

• Points of discontinuity: $x = \frac{\pi}{2} + k\pi$ ($k$ an integer)
• Left limit: $\lim_{x \to (\frac{\pi}{2} + k\pi)^-} \tan x = +\infty$
• Right limit: $\lim_{x \to (\frac{\pi}{2} + k\pi)^+} \tan x = -\infty$
• These are discontinuities of the second kind

The 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

Analysis of the points of discontinuity:

• Points of discontinuity: $x = k\pi$ ($k$ an integer)
• Left limit: $\lim_{x \to (k\pi)^-} \cot x = -\infty$
• Right limit: $\lim_{x \to (k\pi)^+} \cot x = +\infty$
• These are discontinuities of the second kind

The Secant and Cosecant Functions

Secant function: $f(x) = \sec x = \frac{1}{\cos x}$

Properties:

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

Cosecant function: $f(x) = \csc x = \frac{1}{\sin x}$

Properties:

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

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: $f(x) = \sin(x^2)$

Analysis:

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

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

Analysis:

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

Example 3: $f(x) = \tan(x^2)$

Analysis:

• The inner function $g(x) = x^2$ is continuous on $\mathbb{R}$
• The outer function $h(x) = \tan x$ is continuous on its domain
• Therefore $f(x) = h(g(x))$ is continuous on its domain
• Points of discontinuity: those where $x^2 = \frac{\pi}{2} + k\pi$, that is, $x = \pm\sqrt{\frac{\pi}{2} + k\pi}$

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. Asymptotes: there is a vertical asymptote at every point of discontinuity
4. Symmetry: $\sin x$ is an odd function and $\cos x$ is an even function

Applications of Trigonometric Functions

1. Physical Modelling

Trigonometric functions are widely used in physics:

• Vibration: describe simple harmonic motion
• Wave motion: describe various wave phenomena
• Periodic phenomena: describe periodic physical phenomena

2. Engineering Applications

Trigonometric functions have important applications in engineering:

• Signal processing: Fourier analysis
• Circuit analysis: analysis of alternating-current circuits
• Mechanical design: kinematic analysis

Exercises

Exercise 1

Determine the continuity of $f(x) = \sin x$ on $\mathbb{R}$.

Exercise 2

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

Exercise 3

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

Exercise 4

Determine the continuity of $f(x) = \tan(\sin x)$ on $\mathbb{R}$.

Exercise 5

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

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