# Basic trigonometric functions

> IB Mathematics Analysis and Approaches HL · IB Math AA HL 2021+ Syllabus
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-basic-trigonometric-functions/

This sub-topic introduces core periodic trigonometric functions: sine, cosine, tangent, and their reciprocals. You will learn their key graphical features, domain, range, and period, which form the foundation for all further trigonometry in IB AA HL.

**Prerequisites:** [Unit circle definition of trigonometric ratios](https://www.owlsprep.com/study/ib-math-aa-hl-u1-unit-circle-trigonometry/); Basic function properties: domain, range, transformations

## Learning objectives

- Identify domain, range, period and amplitude of basic trigonometric functions
- Sketch and interpret graphs of sine, cosine, tangent and reciprocal functions
- Recall key features of core trigonometric functions for problem solving
- Distinguish between properties of different trigonometric function types

## Sine and Cosine Functions

**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 $2\pi$

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

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

**Worked example:** State the domain, range and period of $f(x) = 3\sin(2x)$

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 $2\pi$, so:
6. $$\text{Period} = \frac{2\pi}{2} = \pi$$

## The Tangent Function

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

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

**Worked example:** Find the location of all vertical asymptotes of $f(x) = \tan(3x)$

1. Parent function $\tan \theta$ has asymptotes when its denominator $\cos \theta = 0$, 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 $f(x)$.

## Reciprocal Trigonometric Functions

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 | $\csc x$ | $\frac{1}{\sin x}$ | $2\pi$ | $x \neq k\pi, k \in \mathbb{Z}$ | $(-\infty, -1] \cup [1, \infty)$ |
| Secant | $\sec x$ | $\frac{1}{\cos x}$ | $2\pi$ | $x \neq \frac{\pi}{2} +k\pi, k \in \mathbb{Z}$ | $(-\infty, -1] \cup [1, \infty)$ |
| Cotangent | $\cot x$ | $\frac{\cos x}{\sin x} = \frac{1}{\tan x}$ | $\pi$ | $x \neq k\pi, k \in \mathbb{Z}$ | $\mathbb{R}$ |

**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. $$\text{Range of } f(x) = (-\infty, -2] \cup [2, \infty)$$

## Graphs of Basic Trigonometric Functions

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.

> **Remembering intercepts at origin**
>
> "Sine Starts at zero, Cosine Starts at one"

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

## Common pitfalls

- **Wrong:** Claiming the period of $\tan(kx)$ is $\frac{2\pi}{k}$
  - Why it fails: Tangent has a fundamental period of $\pi$, not $2\pi$ like sine and cosine
  - Correct: The period of $\tan(kx)$ is always $\frac{\pi}{|k|}$
- **Wrong:** Stating the range of $\csc x$ or $\sec x$ is $[-1, 1]$
  - Why it fails: Reciprocal flips the inequality: values between 0 and 1 become greater than 1, values between -1 and 0 become less than -1
  - Correct: Range of $\csc x$ and $\sec x$ is $(-\infty, -1] \cup [1, \infty)$
- **Wrong:** Including asymptote locations in the domain of trigonometric functions
  - Why it fails: The function is undefined at these points, so they cannot be part of the domain
  - Correct: Exclude all $x$ that make the denominator of the function equal to zero from the domain
- **Wrong:** Assigning an amplitude to tangent, cotangent, secant or cosecant
  - Why it fails: Amplitude is only defined for bounded periodic functions with a maximum and minimum value
  - Correct: Only sine and cosine (and their transformations) have a defined amplitude

## Cheatsheet

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

## 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.

- [Inverse trigonometric functions (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u2-inverse-trigonometric-functions/)
- [Trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-hl-u2-trigonometric-identities/)
- [Function properties: parity and periodicity](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-properties-parity-and-periodicity/)

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