The Six Trig Functions, Amplitude, Period and Graphs

CIE IGCSE Additional Mathematics· 10.1, 10.2, 10.3· 12 min read

1. The Six Trigonometric Functions for Any Angle★★☆☆☆⏱ 3 min

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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.

📘 Definition

Reciprocal Trig Functions

$sec θ = \frac{1}{c o s θ}$ , $cosec θ = \frac{1}{s i n θ}$ , $cot θ = \frac{1}{t a n θ}$

📐 Worked Example

Find the exact values of $sec (12 0^{\circ})$ , $cosec (\frac{3 π}{4})$ and $cot (27 0^{\circ})$ .

1
1. Calculate $cos (12 0^{\circ}) = - \frac{1}{2}$ , so $sec (12 0^{\circ}) = \frac{1}{- \frac{1}{2}} = - 2$
2
1. Calculate $sin (\frac{3 π}{4}) = \frac{2}{2}$ , so $cosec (\frac{3 π}{4}) = \frac{2}{2} = 2$
3
1. Calculate $tan (27 0^{\circ})$ is undefined, so $cot (27 0^{\circ}) = 0$ (since $cot θ = \frac{c o s θ}{s i n θ} = \frac{0}{- 1} = 0$ )

2. Amplitude and Period of Trigonometric Functions★★★☆☆⏱ 3 min

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

📘 Definition

Amplitude and Period Formulas

For $y = a sin (b x) + c$ or $y = a cos (b x) + c$ : Amplitude = $∣ a ∣$ , Period = $\frac{36 0 ^{\circ}}{∣ b ∣}$ (degrees) or $\frac{2 π}{∣ b ∣}$ (radians). For $y = a tan (b x) + c$ : No amplitude, Period = $\frac{18 0 ^{\circ}}{∣ b ∣}$ (degrees) or $\frac{π}{∣ b ∣}$ (radians).

📐 Worked Example

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

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

3. Graphs of Transformed Sine and Cosine Functions★★★☆☆⏱ 4 min

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 $y = c$ ).

📐 Worked Example

Sketch $y = 2 sin (3 x) - 1$ for $0 \leq x \leq 2 π$ , labeling all key features.

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

4. Graphs of Transformed Tangent Functions★★★★☆⏱ 3 min

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

📐 Worked Example

Sketch $y = tan (2 x) + 0.5$ for $0 \leq x \leq π$ , labeling asymptotes and intercepts.

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

5. Common Pitfalls

Wrong move:

Forgetting sec, cosec, cot are reciprocals of cos, sin, tan, not inverse functions

Why:

Confuses reciprocal functions with out-of-scope inverse trig functions, leading to incorrect value calculations

Correct move:

Memorize $sec θ = 1/ cos θ$ , $cosec θ = 1/ sin θ$ , $cot θ = 1/ tan θ$

Wrong move:

Calculating amplitude for tangent functions

Why:

Tangent graphs extend infinitely vertically so they have no defined amplitude

Correct move:

Only calculate amplitude for sine and cosine functions and their transformations

Wrong move:

Using the sin/cos period formula for tan functions

Why:

Sin and cos have period $2 π$ , tan has period $π$ , so the transformed period formulas differ

Correct move:

Use period = $2 π /∣ b ∣$ for sin/cos, $π /∣ b ∣$ for tan (or equivalent degree values)

Wrong move:

Missing asymptote labels on tan graph sketches

Why:

CIE exam markers explicitly deduct marks for unlabeled asymptotes on trig graph questions

Correct move:

Draw dashed lines for asymptotes and label their x-coordinates clearly

Wrong move:

Confusing vertical shift $c$ with amplitude

Why:

Amplitude is the distance from the midline, while $c$ is the position of the midline relative to y=0

Correct move:

First identify the midline $y = c$ , then calculate amplitude as the maximum distance from this midline

6. Quick Reference Cheatsheet

Function	Amplitude	Period (radians)	Period (degrees)	Key Features
$y = a sin (b x) + c$	$∣ a ∣$	$\frac{2 π}{∣ b ∣}$	$\frac{36 0 ^{\circ}}{∣ b ∣}$	Range: $[c - ∣ a ∣, c + ∣ a ∣]$
$y = a cos (b x) + c$	$∣ a ∣$	$\frac{2 π}{∣ b ∣}$	$\frac{36 0 ^{\circ}}{∣ b ∣}$	Range: $[c - ∣ a ∣, c + ∣ a ∣]$
$y = a tan (b x) + c$	None	$\frac{π}{∣ b ∣}$	$\frac{18 0 ^{\circ}}{∣ b ∣}$	Asymptotes at $b x = \frac{π}{2} + k π$
$sec θ$	None	$2 π$	$36 0^{\circ}$	Undefined where $cos θ = 0$
$cosec θ$	None	$2 π$	$36 0^{\circ}$	Undefined where $sin θ = 0$
$cot θ$	None	$π$	$18 0^{\circ}$	Undefined where $tan θ = 0$

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.