Basic Trigonometry

Revise Basic Trigonometry for Pure Mathematics 1 WMA11 (AS Level) — revision notes and instant AI marking.

📖 Revision notes · preview

Chapter Overview

Key Concepts at a Glance

The three trig ratios — sin, cos, tan — relate an angle to the side lengths of a right-angled triangle via SOHCAHTOA.
The unit circle (radius = 1, centre = origin) extends trig to angles beyond 90°. Any point on it has coordinates (cos θ, sin θ).
To find a missing side in a right-angled triangle: choose the right SOHCAHTOA ratio, substitute, and rearrange.
To find a missing angle in a right-angled triangle: use the inverse trig function — sin⁻¹, cos⁻¹, or tan⁻¹.
For non-right-angled triangles, use the Sine Rule (when you have opposite pairs), the Cosine Rule (when you have an angle between two sides, or all three sides), or the Area Formula ½ab sin C.
The Sine Rule always returns the acute angle; if you need the obtuse one, subtract from 180°.
These rules are NOT in the formula booklet — you must memorise them all.

Section 1

Definitions of Trigonometric Functions

What is Trigonometry?

The word itself comes from the Greek trigonon (triangle) + metron (measure). At its heart, trigonometry answers the question: "If I know one angle and one side of a triangle, what are all the other sides?"

Every time you use sin, cos, or tan, you're using a ratio — a fraction that compares two sides of a right-angled triangle. The amazing thing is that this ratio depends only on the angle, not on how big the triangle is. A 30° angle always produces the same ratio, whether your triangle is the size of a stamp or the size of a football pitch.

Labelling the Sides

Before you can write any trig ratio, you must label the three sides relative to the angle you're working with:

Hypotenuse — always the longest side; always opposite the right angle.

Opposite — the side directly across from your angle θ.

Adjacent — the side sandwiched between your angle θ and the right angle.

Right-Angled Triangle — Side Labels

Key Insight

"Adjacent" and "Opposite" change depending on which angle you call θ. The Hypotenuse always stays the same (longest side, opposite the right angle). Relabel the triangle if you switch to a different angle.

The Three Ratios — SOHCAHTOA

SOHCAHTOA is the mnemonic that packs three formulas into one memorable word:

Letters	Function	Ratio	In words
SOH	Sin θ	sin θ = O / H	Opposite over Hypotenuse
CAH	Cos θ	cos θ = A / H	Adjacent over Hypotenuse
TOA	Tan θ	tan θ = O / A	Opposite over Adjacent

Finding a Missing Angle — Inverse Functions

θ = sin⁻¹(O/H) θ = cos⁻¹(A/H) θ = tan⁻¹(O/A)

When you know two sides and want the angle, hit the sin/cos/tan button with the ⁻¹ (inverse) on your calculator. This "undoes" the trig function.

Common Mistake

Students often write a correct ratio like sin 30 = x/10 and then forget to multiply both sides — they just write x = sin 30 and stop. Always rearrange fully: x = 10 × sin 30 = 5.

The Unit Circle

radius 1

Section 2

Right-Angled Triangles

How to Use SOHCAHTOA — Step by Step

Follow this procedure every single time — it eliminates almost all errors:

↓

🔓 Read the full Basic Trigonometry note → You're seeing the preview · sign in to read it all

Also in the full note

Non-Right-Angled Triangles
What to Memorise
Concepts Checklist
Exam Tips & Common Traps
Worked Example — Finding Two Unknowns
The Labelling Convention
The Cosine Rule
Area of Any Triangle

What's inside

📖 Revision notes 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

📄 Practise Basic Trigonometry with Pure Mathematics 1 WMA11 past papers Every paper with its mark scheme — answer online, marked instantly. Open →

Read the full Basic Trigonometry notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →