Trigonometry & Euclidean Geometry – Solving the World One Angle at a Time

Admin
Jul 27
2 min read

Dear Grade 12s

Let’s face it — when many learners hear “trigonometry” or “geometry,” they feel nervous. And I get it.

These topics can feel abstract, filled with rules and angles that seem to go in circles (literally and figuratively!).

But here’s the truth: you can master them.

These are not just mathematical topics — they’re about learning how to solve problems logically, to see patterns, and to trust your working process.

Let’s walk through them together — one diagram, one identity, one proof at a time.

📘 Part 1: Trigonometry – The Language of Angles and Triangles

Trigonometry is about the relationships between the angles and sides of triangles, especially right-angled ones.

🧮 1. The Basic Ratios (SOHCAHTOA)

In a right-angled triangle:

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

✅ Use these for basic triangle problems and to find missing sides or angles.

🧠 2. Special Angles and the Unit Circle

Learn the values of sine, cosine, and tangent for angles like 0°, 30°, 45°, 60°, and 90°.

The unit circle helps with:

Understanding signs in different quadrants
Working with angles greater than 90°
Using CAST diagram (Cosine, All, Sine, Tangent positive in Quadrants IV, I, II, III)

🧩 3. Trig Identities & Equations

You must know and apply these identities:

sin²θ + cos²θ = 1
tanθ = sinθ / cosθ

Solve trig equations by:

Simplifying
Using identities
Solving for general solutions (using reference angles)

📐 4. Area and Rules in Non-Right Triangles

When the triangle isn't right-angled:

Use Sine Rule:a/sinA = b/sinB = c/sinC
Use Cosine Rule:c² = a² + b² – 2ab cosC
Area of triangle:A = ½ ab sinC

✅ These rules are key for real-life applications like surveying and navigation.

📘 Part 2: Euclidean Geometry – The Power of Proof

Euclidean Geometry is the art of logical reasoning using known theorems and definitions.

📚 1. Important Theorems to Know:

Angles in the same segment are equal
Angle at the centre = 2 × angle at the circumference
Opposite angles of a cyclic quadrilateral are supplementary
Exterior angle of triangle = sum of two opposite interior angles

✅ You’ll be expected to use logical steps with clear reasons.

🔍 2. Tips for Proof and Reasoning:

Write neatly and label clearly
Use correct reasons from the geometry rule book (no guessing!)
When stuck:
- Look for isosceles triangles, parallel lines, or cyclic quads
- Mark equal angles and sides on the diagram
- Go step by step — don’t rush

📝 Common Question Types:

Prove this angle is equal to that angle
Show that a shape is a cyclic quadrilateral
Prove that two lines are parallel or equal

✅ Don’t just answer — justify each step!

🧠 Exam Tips:

📌 Practise labelled diagrams — don’t just try to “read” them

📌 Use a ruler, pencil, and compass when needed

📌 Learn the common reasons used in geometry proofs

📌 Revise past papers — similar diagrams appear often

📌 For trig: Know when to use rules, identities, or ratios