Trigonometric Functions

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Edexcel IAL Maths · Pure 1

Trigonometric Functions

Graphs, properties, transformations — everything in one place.

The Big Idea: Sin, cos, and tan produce repeating, predictable wave-like graphs with specific symmetries — and every transformation rule you already know for regular functions works on them too.

Chapter Overview

1Sin, cos and tan each have a unique periodic graph shape you must be able to sketch from memory.
2Sin x has rotational symmetry about the origin; cos x is symmetric about the y-axis.
3Tan x has asymptotes (undefined values) at ±90°, ±270°, ±450°… and repeats every 180°.
4To solve trig equations in a range, always sketch the graph — your calculator only gives the principal value.
5Vertical stretch: y = n·f(x) multiplies every y-value by n, x-values unchanged.
6Horizontal squash: y = f(nx) multiplies every x-value by 1/n, y-values unchanged.
7Horizontal translation: y = f(x+c) shifts the graph left by c (if c > 0).
8Reflection: negative outside (y = −f(x)) flips in x-axis; negative inside (y = f(−x)) flips in y-axis.

Graphs of Trigonometric Functions

y = sin x & y = cos x

The Sine and Cosine Waves

Think of sin x and cos x as the same wave — cos x is just sin x shifted 90° to the left. They share the same height, the same shape, and the same period.

y = sin x (red) · y = cos x (blue) · Range: −1 to 1 · Period: 360°

Property	y = sin x	y = cos x
Range	−1 to 1	−1 to 1
Period	360°	360°
At x = 0	Passes through origin (0, 0)	Passes through (0, 1)
Symmetry	Rotational about origin → sin(−x) = −sin(x)	Symmetric about y-axis → cos(−x) = cos(x)
Maxima	x = 90°, 450°, … (and −270°, …)	x = 0°, 360°, … (and −360°, …)
Zeros (positive)	x = 0°, 180°, 360°, …	x = 90°, 270°, …

🧠 The Key Difference

Sin starts at 0 and rises — like a wave leaving shore. Cos starts at its peak (1) and falls — like a wave that's already crested. They're the same wave, just starting at different points.

y = tan x

The Tangent Function

Tan is the odd one out. Unlike sin and cos, it has no maximum or minimum — it shoots from negative infinity to positive infinity, then resets. This happens because tan x = sin x / cos x, and whenever cos x = 0 (at 90°, 270°, etc.), you'd be dividing by zero — which is undefined.

y = tan x — asymptotes (dashed red) at ±90°, ±270°, … · Period: 180°

Property	y = tan x
Range	−∞ to +∞ (all real numbers)
Period	180° (not 360°!)
Asymptotes	x = ±90°, ±270°, ±450°, … (wherever cos x = 0)
At x = 0	Passes through origin (0, 0)
Zeros	x = 0°, ±180°, ±360°, …
Symmetry	Rotational about origin → tan(−x) = −tan(x)

⚠️ Classic Mistake

Students often say tan x has a period of 360°. It doesn't — it's 180°. The pattern repeats twice as fast as sin and cos.

Using Trig Graphs to Solve Equations

Finding All Solutions in a Given Range

principal value

1

Identify the function and range. E.g. for sin x = −0.25, the function is sin and the range might be −180° ≤ x ≤ 270°.

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Also in the full note

Transformations of Trigonometric Functions
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
y = n · f(x)
y = f(nx) — Changing the Period
y = f(x + c) — Shifting Left or Right
y = f(x) + d | y = −f(x) | y = f(−x)

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Trigonometric Functions