An Introduction to Trigonometry
Date: 2025-06-11 • Author: Trafalgar Tuition
Tags: GCSE, Maths, FTSE, Trigonometry, Proof
Introduction
When someone asks, “What does sine or cosine mean?”, the usual answer is “something on the calculator.” But these functions describe something much deeper — the relationship between sides and angles in right-angled triangles.
You might have had SOHCAHTOA drilled into you at school, but do you really understand it?
The Basics
In any right-angled triangle, we label the sides relative to a chosen angle \( \theta \):
- Hypotenuse — the longest side (always opposite the right angle)
- Opposite — the side directly opposite the angle \( \theta \)
- Adjacent — the side next to \( \theta \) that’s not the hypotenuse
From this, we define the three core trigonometric ratios:
$$ \sin \theta = \frac{\text{Opp}}{\text{Hyp}}, \quad \cos \theta = \frac{\text{Adj}}{\text{Hyp}}, \quad \tan \theta = \frac{\text{Opp}}{\text{Adj}} $$
(Insert diagram here showing triangle with angle \( \theta \) and labelled sides)
Beyond Right-Angled Triangles
These definitions only apply to right-angled triangles. For other types — scalene, obtuse, equilateral — we use broader tools like the sine rule and cosine rule (covered in later posts).
What About SOHCAHTOA?
The mnemonic SOHCAHTOA can help some students remember the ratios — but it often leads to confusion when it’s applied blindly. Students forget to label sides correctly or pick the wrong ratio entirely.
That’s why I prefer to teach trigonometry by focusing on the relationships they represent— not just the letters.
Conclusion
Trigonometry underpins much of GCSE and A-level maths — and it all starts from these definitions.
🔗 Next up: After the introduction of SOHCAHTOA, the next stop is another trigonometry equation, the sine rule, which lets us handle any type of triangle, not simply the right-angled ones.