Basic trigonometric functions

IB Mathematics Analysis and Approaches HL· Unit 2: Functions, Topic 2.8· 15 min read

1. Sine and Cosine Functions★★☆☆☆⏱ 5 min

📘 Definition

Periodic Function

A function $f (x)$ is periodic if there exists a constant $T > 0$ such that $f (x + T) = f (x)$ for all $x$ in the domain. The smallest such $T$ is the fundamental period of the function.

Example:

$f (x) = sin x$ has fundamental period $2 π$

Sine and cosine are defined for all real $x$ using the unit circle: $sin θ$ is the $y$ -coordinate and $cos θ$ is the $x$ -coordinate of the point on the unit circle at angle $θ$ from the positive $x$ -axis.

Domain: All real numbers $x \in R$ for both functions
Range: $[- 1, 1]$ for both functions
Fundamental period: $2 π$ for both
Amplitude: $1$ for both

📐 Worked Example

State the domain, range and period of $f (x) = 3 sin (2 x)$

1
Vertical scaling does not change the domain of sine, since sine is defined for all real arguments. Domain is unchanged from the parent function:
2
$\text{Domain: } x \in \mathbb{R}$
3
Original sine has range $[- 1, 1]$ . Scaling all output values by 3 gives the new range:
4
$\text{Range: } [3 \times (-1), 3 \times 1] = [-3, 3]$
5
Horizontal scaling by factor $\frac{1}{2}$ halves the period of the parent function. Original period is $2 π$ , so:
6
$\text{Period} = \frac{2\pi}{2} = \pi$

2. The Tangent Function★★☆☆☆⏱ 5 min

Unlike sine and cosine, tangent is defined as the ratio $tan θ = \frac{s i n θ}{c o s θ}$ , so it is undefined when $cos θ = 0$ . This creates vertical asymptotes at these undefined points.

📘 Definition

Key Features of $\tan x$

Tangent has a fundamentally different period and range from sine and cosine, because it is a ratio of two periodic functions.

Domain: All real $x$ except $x = \frac{π}{2} + k π, k \in Z$
Range: All real numbers $R$
Fundamental period: $π$ (not $2 π$ !)
No defined amplitude (range is unbounded)

📐 Worked Example

Find the location of all vertical asymptotes of $f (x) = tan (3 x)$

1
Parent function $tan θ$ has asymptotes when its denominator $cos θ = 0$ , which occurs at:
2
$\theta = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$
3
Set the argument of our transformed tangent equal to this condition, then solve for $x$ :
4
$3x = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{6} + \frac{k\pi}{3}, \quad k \in \mathbb{Z}$
5
These are all locations of the vertical asymptotes of $f (x)$ .

3. Reciprocal Trigonometric Functions★★★☆☆⏱ 4 min

The reciprocals of sine, cosine and tangent give three additional core trigonometric functions: cosecant, secant and cotangent. Each reciprocal inherits the periodicity of the original function, but has vertical asymptotes where the original function equals zero.

Name	Notation	Definition	Period	Domain	Range
Cosecant	$csc x$	$\frac{1}{s i n x}$	$2 π$	$x \neq = k π, k \in Z$	$(- \infty, - 1] \cup [1, \infty)$
Secant	$sec x$	$\frac{1}{c o s x}$	$2 π$	$x \neq = \frac{π}{2} + k π, k \in Z$	$(- \infty, - 1] \cup [1, \infty)$
Cotangent	$cot x$	$\frac{c o s x}{s i n x} = \frac{1}{t a n x}$	$π$	$x \neq = k π, k \in Z$	$R$

📐 Worked Example

Find the range of $f (x) = 2 sec x$

1
Start with the range of the parent function $sec x$ , which we know from the table above:
2
$\text{Range of } \sec x = (-\infty, -1] \cup [1, \infty)$
3
Vertical scaling by 2 multiplies all output values by 2, so we scale each interval of the range:
4
$2(-\infty, -1] = (-\infty, -2], \quad 2[1, \infty) = [2, \infty)$
5
The combined range is therefore:
6
$\text{Range of } f(x) = (-\infty, -2] \cup [2, \infty)$

4. Graphs of Basic Trigonometric Functions★★☆☆☆⏱ 5 min

All basic trigonometric functions have characteristic shapes you need to recognize and sketch quickly for IB exam questions. You must be able to mark key intercepts, maxima/minima and asymptotes correctly.

Sine crosses $(0, 0)$ , has maximum at $(\frac{π}{2}, 1)$ , zero at $(π, 0)$ , minimum at $(\frac{3 π}{2}, - 1)$ , zero at $(2 π, 0)$
Cosine crosses $(0, 1)$ , zero at $(\frac{π}{2}, 0)$ , minimum at $(π, - 1)$ , zero at $(\frac{3 π}{2}, 0)$ , maximum at $(2 π, 1)$
Tangent crosses $(0, 0)$ , has asymptotes at $\pm \frac{π}{2}$ , and increases continuously from $- \infty$ to $+ \infty$ between any two consecutive asymptotes

5. Common Pitfalls

Wrong move:

Claiming the period of $tan (k x)$ is $\frac{2 π}{k}$

Why:

Tangent has a fundamental period of $π$ , not $2 π$ like sine and cosine

Correct move:

The period of $tan (k x)$ is always $\frac{π}{∣ k ∣}$

Wrong move:

Stating the range of $csc x$ or $sec x$ is $[- 1, 1]$

Why:

Reciprocal flips the inequality: values between 0 and 1 become greater than 1, values between -1 and 0 become less than -1

Correct move:

Range of $csc x$ and $sec x$ is $(- \infty, - 1] \cup [1, \infty)$

Wrong move:

Including asymptote locations in the domain of trigonometric functions

Why:

The function is undefined at these points, so they cannot be part of the domain

Correct move:

Exclude all $x$ that make the denominator of the function equal to zero from the domain

Wrong move:

Assigning an amplitude to tangent, cotangent, secant or cosecant

Why:

Amplitude is only defined for bounded periodic functions with a maximum and minimum value

Correct move:

Only sine and cosine (and their transformations) have a defined amplitude

6. Quick Reference Cheatsheet

Function	Domain	Range	Period	Amplitude
$sin x$	$R$	$[- 1, 1]$	$2 π$	$1$
$cos x$	$R$	$[- 1, 1]$	$2 π$	$1$
$tan x$	$x \neq = \frac{π}{2} + k π$	$R$	$π$	None
$csc x$	$x \neq = k π$	$(- \infty, - 1] \cup [1, \infty)$	$2 π$	None
$sec x$	$x \neq = \frac{π}{2} + k π$	$(- \infty, - 1] \cup [1, \infty)$	$2 π$	None
$cot x$	$x \neq = k π$	$R$	$π$	None

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find range of transformed secant function
2022 · 1
Identify asymptotes of tangent function
2023 · 2
Sketch graph of basic sine function

What's Next

Basic trigonometric functions are the foundation for all further work in trigonometry in IB AA HL. Next, you will learn how to transform these functions with translations, reflections and scalings, which allows you to model periodic real-world phenomena such as tidal motion or oscillating springs. You will also use these basic functions to solve more complex trigonometric equations and prove trigonometric identities, which are frequent long questions in both Paper 1 and Paper 2 of the IB exam.