Periodic and Trigonometric Functions
- Calculus 1.3
Periodic Functions
A periodic function is a function that repeats its values at regular intervals. The smallest positive value of $p$ such that $f(x + p) = f(x)$ for all $x$ in the domain of $f$ is called the period of the function. (Sometimes, the word “smallest” is omitted) The period of a periodic function is the length of one complete cycle of the function. Trigonometric functions are the most common examples of periodic functions.
Trigonometric Functions
Angle Measurement
Angles are measured in radians. Radians are defined as the ratio of the length of the arc subtended by the angle to the radius of the circle.
\[ \theta = \frac{s}{r} \]
where $\theta$ is the angle in radians, $s$ is the length of the arc, and $r$ is the radius of the circle. The relationship between degrees and radians is given by:
\[ \pi \text{ rad} = 180^\circ \]
Sine and Cosine Functions
The sine and cosine functions can be defined in several ways, typically using the unit circle(geometrical way) or power series(analytical way). It is convenient to define them using the power series for latter discussions such as differentiation, but here we will use the unit circle definition for simplicity.
The unit circle is a circle with a radius of 1 centered at the origin of the Cartesian coordinate system. The coordinates of a point on the unit circle can be expressed in terms of the angle $\theta$ as follows, and thus the sine and cosine functions are defined as:
\[ x = \cos(\theta), \quad y = \sin(\theta) \]
Sine and cosine functions on the unit circle
The sine and cosine functions are periodic with a period of $2\pi$.
The Six Basic Trigonometric Functions
The six basic trigonometric functions are defined as follows:
| Sine | Cosine | Tangent | Cosecant | Secant | Cotangent | |
|---|---|---|---|---|---|---|
| Functions | $\sin x$ | $\cos x$ | $\tan x = \dfrac{\sin x}{\cos x}$ | $\csc x = \dfrac{1}{\sin x}$ | $\sec x = \dfrac{1}{\cos x}$ | $\cot x = \dfrac{\cos x}{\sin x}$ |
| Domain | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R} \setminus \left( \mathbb{Z} + \dfrac{1}{2} \right)\pi$ | $\mathbb{R} \setminus \mathbb{Z}\pi$ | $\mathbb{R} \setminus \left( \mathbb{Z} + \dfrac{1}{2} \right)\pi$ | $\mathbb{R} \setminus \mathbb{Z}\pi$ |
| Range | $[-1, 1]$ | $[-1, 1]$ | $\mathbb{R}$ | $\mathbb{R} \setminus (-1,1)$ | $\mathbb{R} \setminus (-1,1)$ | $\mathbb{R}$ |
| Period | $2\pi$ | $2\pi$ | $\pi$ | $2\pi$ | $2\pi$ | $\pi$ |
| Symmetry | Odd | Even | Odd | Odd | Even | Odd |
Animation of the six basic trigonometric functions
Graph of the six basic trigonometric functions
Trigonometrical Identities
Pythagorean Identities
\[ \sin^2 x + \cos^2 x = 1 \nl \sec^2 x = 1 + \tan^2 x \nl \csc^2 x = 1 + \cot^2 x \]
Shift Identities
\[ \sin\left(x \pm \pi/2\right) = \pm \cos x \quad \cos\left(x \pm \pi/2\right) = \mp \sin x \nl \sin(x+ \pi) = -\sin x \quad \cos(x + \pi) = -\cos x \]
Angle Addition Identities
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \nl \cos(x + y) = \cos x \cos y - \sin x \sin y \nl \tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \]
Proof is replaced by the following image:
The Law of Cosines
If $a,b,c$ are the lengths of the sides of a triangle opposite to angles $A,B,C$, respectively, then:
\[ c^2 = a^2 + b^2 - 2ab \cos C \nl a^2 = b^2 + c^2 - 2bc \cos A \nl b^2 = c^2 + a^2 - 2ca \cos B \]
Proof
It can be proven by combining following identities:
\[ c = a \cos B + b \cos A \nl a = b \cos C + c \cos B \nl b = c \cos A + a \cos C \]
Extra Properties
\[ \abs{ \sin \theta } \leq \abs{ \theta } \leq \abs{ \tan \theta } \quad \left( \abs{ \theta } < \frac{\pi}{2} \right) \nt 0\leq 1 - \cos \theta \leq \frac{ \theta^2 }{2} \]
These are very easy to prove, so I will not include the proofs here.
