# The Six Trig Functions, Amplitude, Period and Graphs

> CIE IGCSE Additional Mathematics · 2025-2027
> Source: https://www.owlsprep.com/study/cie-0606-u10-the-six-trig-functions-amplitude/

This guide covers all six trigonometric functions, amplitude, period, and transformations of sin, cos, tan graphs required for CIE IGCSE Additional Mathematics 0606. You will learn to sketch graphs and identify their key features for exam questions.

**Prerequisites:** Basic trigonometric ratios for right triangles; Radian measure and unit circle

## Learning objectives

- Define the six trigonometric functions (sin, cos, tan, sec, cosec, cot) for angles of any magnitude in degrees or radians
- Calculate amplitude and period for sine, cosine and tangent functions and their transformations
- Sketch graphs of $a \sin(bx)+c$, $a \cos(bx)+c$, $a \tan(bx)+c$, including asymptotes for tangent
- Identify key features of trigonometric graphs from their equations

## The Six Trigonometric Functions for Any Angle

The six trigonometric functions are defined for angles of any magnitude (positive, negative, degrees or radians) using the unit circle. The three reciprocal functions (sec, cosec, cot) are derived from the core sin, cos, tan functions.

**Reciprocal Trig Functions** — $\sec \theta = \frac{1}{\cos \theta}$, $\cosec \theta = \frac{1}{\sin \theta}$, $\cot \theta = \frac{1}{\tan \theta}$

**Worked example:** Find the exact values of $\sec(120^\circ)$, $\cosec\left(\frac{3\pi}{4}\right)$ and $\cot(270^\circ)$.

1. 1. Calculate $\cos(120^\circ) = -\frac{1}{2}$, so $\sec(120^\circ) = \frac{1}{-\frac{1}{2}} = -2$
2. 2. Calculate $\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$, so $\cosec\left(\frac{3\pi}{4}\right) = \frac{2}{\sqrt{2}} = \sqrt{2}$
3. 3. Calculate $\tan(270^\circ)$ is undefined, so $\cot(270^\circ) = 0$ (since $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{0}{-1} = 0$)

*Calculator:* forbidden

## Amplitude and Period of Trigonometric Functions

Amplitude only applies to sine and cosine functions, which have bounded ranges. All six trig functions are periodic, repeating their values at regular intervals.

**Amplitude and Period Formulas** — For $y = a \sin(bx) + c$ or $y = a \cos(bx) + c$: Amplitude = $|a|$, Period = $\frac{360^\circ}{|b|}$ (degrees) or $\frac{2\pi}{|b|}$ (radians). For $y = a \tan(bx) + c$: No amplitude, Period = $\frac{180^\circ}{|b|}$ (degrees) or $\frac{\pi}{|b|}$ (radians).

**Worked example:** Find the amplitude and period (in radians) of $y = 3\cos(2x) + 1$ and $y = 2\tan(4x) - 3$.

1. 1. For $y = 3\cos(2x) +1$: $a=3$, $b=2$, so Amplitude = $|3| = 3$, Period = $\frac{2\pi}{2} = \pi$
2. 2. For $y = 2\tan(4x) -3$: No amplitude, $b=4$, so Period = $\frac{\pi}{4}$

*Calculator:* forbidden

## Graphs of Transformed Sine and Cosine Functions

For transformed sin and cos graphs, $a$ controls amplitude, $b$ scales the period, and $c$ shifts the entire graph vertically by $c$ units (midline is $y=c$).

> **Exam tip**
>
> Always label the midline, maximum, minimum, and x-intercepts on sin/cos graph sketches for full marks.

**Worked example:** Sketch $y = 2\sin(3x) - 1$ for $0 \leq x \leq 2\pi$, labeling all key features.

1. 1. Identify key values: Amplitude = 2, Midline $y=-1$, Period = $\frac{2\pi}{3}$
2. 2. Maximum value: $-1 + 2 = 1$, Minimum value: $-1 -2 = -3$
3. 3. Plot 3 full cycles over $0 \leq x \leq 2\pi$, mark peaks at $y=1$, troughs at $y=-3$, midline crossing points at $y=-1$

## Graphs of Transformed Tangent Functions

Tangent graphs have no amplitude, and feature vertical asymptotes where the function is undefined. The transformation $a \tan(bx) + c$ scales the vertical stretch by $a$, scales the period by $\frac{1}{b}$, and shifts vertically by $c$.

> **Exam tip**
>
> CIE examiners require dashed lines for asymptotes and clear x-coordinate labels for asymptotes; missing these will cost 1-2 marks per sketch.

**Worked example:** Sketch $y = \tan(2x) + 0.5$ for $0 \leq x \leq \pi$, labeling asymptotes and intercepts.

1. 1. Period = $\frac{\pi}{2}$, so 2 full cycles over $0 \leq x \leq \pi$
2. 2. Asymptotes occur where $2x = \frac{\pi}{2} + k\pi$, so $x = \frac{\pi}{4}, \frac{3\pi}{4}$ for the given domain
3. 3. Midline is $y=0.5$, plot the graph crossing the midline at $x=0, \frac{\pi}{2}, \pi$, approaching each asymptote without touching it

## Common pitfalls

- **Wrong:** Forgetting sec, cosec, cot are reciprocals of cos, sin, tan, not inverse functions
  - Why it fails: Confuses reciprocal functions with out-of-scope inverse trig functions, leading to incorrect value calculations
  - Correct: Memorize $\sec \theta = 1/\cos \theta$, $\cosec \theta = 1/\sin \theta$, $\cot \theta = 1/\tan \theta$
- **Wrong:** Calculating amplitude for tangent functions
  - Why it fails: Tangent graphs extend infinitely vertically so they have no defined amplitude
  - Correct: Only calculate amplitude for sine and cosine functions and their transformations
- **Wrong:** Using the sin/cos period formula for tan functions
  - Why it fails: Sin and cos have period $2\pi$, tan has period $\pi$, so the transformed period formulas differ
  - Correct: Use period = $2\pi/|b|$ for sin/cos, $\pi/|b|$ for tan (or equivalent degree values)
- **Wrong:** Missing asymptote labels on tan graph sketches
  - Why it fails: CIE exam markers explicitly deduct marks for unlabeled asymptotes on trig graph questions
  - Correct: Draw dashed lines for asymptotes and label their x-coordinates clearly
- **Wrong:** Confusing vertical shift $c$ with amplitude
  - Why it fails: Amplitude is the distance from the midline, while $c$ is the position of the midline relative to y=0
  - Correct: First identify the midline $y=c$, then calculate amplitude as the maximum distance from this midline

## Cheatsheet

| Function | Amplitude | Period (radians) | Period (degrees) | Key Features |
| --- | --- | --- | --- | --- |
| $y = a \sin(bx) + c$ | $\|a\|$ | $\frac{2\pi}{\|b\|}$ | $\frac{360^\circ}{\|b\|}$ | Range: $[c-\|a\|, c+\|a\|]$ |
| $y = a \cos(bx) + c$ | $\|a\|$ | $\frac{2\pi}{\|b\|}$ | $\frac{360^\circ}{\|b\|}$ | Range: $[c-\|a\|, c+\|a\|]$ |
| $y = a \tan(bx) + c$ | None | $\frac{\pi}{\|b\|}$ | $\frac{180^\circ}{\|b\|}$ | Asymptotes at $bx = \frac{\pi}{2} + k\pi$ |
| $\sec \theta$ | None | $2\pi$ | $360^\circ$ | Undefined where $\cos \theta = 0$ |
| $\cosec \theta$ | None | $2\pi$ | $360^\circ$ | Undefined where $\sin \theta = 0$ |
| $\cot \theta$ | None | $\pi$ | $180^\circ$ | Undefined where $\tan \theta = 0$ |

## What's next

Now that you have mastered the six trig functions, amplitude, period and their graphs, you are ready to move on to solving trigonometric equations and applying these concepts to calculus problems in the CIE IGCSE Additional Mathematics 0606 syllabus. You will use your knowledge of graph key features to find solutions to equations involving multiple trig functions in given domains, and later differentiate and integrate trigonometric functions for kinematics and area under curve questions. Ensure you practice sketching a variety of transformed trig graphs regularly, as these are frequently tested in both calculator and non-calculator papers to assess your understanding of function transformations and trigonometric properties.

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