Review of Trigonometric Functions
The trigonometric functions that form the foundation of Fourier analysis
Introduction (High School Level)
Introduction
Fourier analysis is the technique of expressing a complicated function as a superposition of sinusoidal waves (sine and cosine waves). A solid understanding of the trigonometric functions is therefore essential.
This chapter reviews the definitions, graphs, and important identities of the trigonometric functions.
Note: For a more detailed treatment of the trigonometric functions, see Geometry Introduction. In particular, Chapter 1: Trigonometric Ratios and Chapter 2: Trigonometric Functions provide a systematic treatment ranging from the right-triangle definition to general angles, radian measure, and the properties of the graphs.
Definition of the Trigonometric Functions
Definition via the unit circle
Consider a point $P$ on the unit circle (the circle of radius $1$ centred at the origin). Let $P = (x, y)$ be the point obtained by rotating counter-clockwise from the positive $x$-axis through an angle $\theta$. Then:
$$\cos\theta \triangleq x, \quad \sin\theta \triangleq y.$$
The tangent is defined by:
$$\tan\theta \triangleq \dfrac{y}{x} = \dfrac{\sin\theta}{\cos\theta} \quad (\cos\theta \neq 0).$$
Basic values
The positions on the unit circle for several characteristic angles are shown below.
| $\theta$ | $0$ | $\dfrac{\pi}{6}$ | $\dfrac{\pi}{4}$ | $\dfrac{\pi}{3}$ | $\dfrac{\pi}{2}$ | $\pi$ |
|---|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $1$ | $0$ |
| $\cos\theta$ | $1$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{1}{2}$ | $0$ | $-1$ |
Graphs
The sine function $y = \sin x$
- Period: $2\pi$
- Amplitude: $1$ (maximum $1$, minimum $-1$)
- Odd function passing through the origin: $\sin(-x) = -\sin x$
The cosine function $y = \cos x$
- Period: $2\pi$
- Amplitude: $1$ (maximum $1$, minimum $-1$)
- Even function symmetric about the $y$-axis: $\cos(-x) = \cos x$
Relationship between sine and cosine
The cosine function is the sine function shifted to the left by $\pi/2$:
$$\cos x = \sin\left(x + \dfrac{\pi}{2}\right).$$
Important Identities
The Pythagorean identity (proof)
$$\sin^2\theta + \cos^2\theta = 1.$$
The addition formulas (proof)
$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta,$$
$$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta.$$
Product-to-sum formulas (proof)
$$\sin\alpha\cos\beta = \dfrac{1}{2}\{\sin(\alpha+\beta) + \sin(\alpha-\beta)\},$$
$$\cos\alpha\cos\beta = \dfrac{1}{2}\{\cos(\alpha-\beta) + \cos(\alpha+\beta)\},$$
$$\sin\alpha\sin\beta = \dfrac{1}{2}\{\cos(\alpha-\beta) - \cos(\alpha+\beta)\}.$$
Orthogonality (the heart of Fourier analysis)
For integers $m, n$, the following integrals over $[0, 2\pi]$ hold:
$$\int_0^{2\pi} \sin(mx)\cos(nx)\,dx = 0,$$
$$\int_0^{2\pi} \sin(mx)\sin(nx)\,dx = \begin{cases} 0 & (m \neq n) \\ \pi & (m = n \neq 0) \end{cases},$$
$$\int_0^{2\pi} \cos(mx)\cos(nx)\,dx = \begin{cases} 0 & (m \neq n) \\ \pi & (m = n \neq 0) \\ 2\pi & (m = n = 0) \end{cases}.$$
Just as two vectors are said to be "orthogonal" when their inner product is $0$, two functions are said to be "orthogonal" when the integral of their product is $0$. Thanks to this orthogonality, one can extract particular frequency components from a complicated wave, and this is the key tool for computing Fourier coefficients.
Summary
- The trigonometric functions $\sin$ and $\cos$ are defined as the coordinates of points on the unit circle.
- $\sin x$ is an odd function and $\cos x$ is an even function.
- Both have period $2\pi$.
- The product-to-sum formulas and the orthogonality relations form the foundation of Fourier analysis.
From the next chapter onward, we will study how to represent various functions by superposing sinusoidal waves $A\sin(k x + \varphi)$ with different amplitudes, periods, and phases.
Frequently Asked Questions
Q1. What are the basic properties of the trigonometric functions?
A: The functions $\sin x$ and $\cos x$ are periodic with period $2\pi$, and they satisfy $\sin^2 x + \cos^2 x = 1$ and $\sin(x+y) = \sin x \cos y + \cos x \sin y$ (the addition formula). Fourier analysis aims to express every periodic function as a superposition of trigonometric functions.
Q2. What is the orthogonality of trigonometric functions?
A: On the interval $[0, 2\pi]$, the relations $\int_0^{2\pi} \sin(nx)\cos(mx)\,dx = 0$ and $\int_0^{2\pi} \sin(nx)\sin(mx)\,dx = \pi\delta_{nm}$ (for $n, m \geq 1$) hold. This is called the orthogonality of the trigonometric system and is the foundation for computing Fourier coefficients.
Q3. Why are trigonometric functions important in Fourier analysis?
A: Because they form a basis for periodic functions. The set $\{1, \cos x, \sin x, \cos 2x, \sin 2x, \ldots\}$ is a complete orthonormal system in $L^2[0,2\pi]$, and every square-integrable periodic function can be expressed as a linear combination of these functions (its Fourier expansion).