Cambridge Syllabus Notes

Geometry – Trigonometry & Pythagoras

What you’ll learn

Basic trigonometric ratios: sine, cosine and tangent
The Pythagorean theorem – the “magic” of right‑angled triangles
Using the sine rule and cosine rule to solve any triangle
Exam‑ready tips and tricks

Sine, Cosine & Tangent – the “Ratios” of a Right Triangle

Imagine a right‑angled triangle as a ladder leaning against a wall. The ladder’s length is the hypotenuse (the longest side). The height of the wall is the opposite side to the angle you’re interested in, and the distance from the wall to the base of the ladder is the adjacent side.

Ratio	Formula	Example
$\sin \theta$	$\displaystyle \frac{\text{opposite}}{\text{hypotenuse}}$	If opposite = 3 cm, hypotenuse = 5 cm → $\sin \theta = \frac{3}{5}=0.6$
$\cos \theta$	$\displaystyle \frac{\text{adjacent}}{\text{hypotenuse}}$	If adjacent = 4 cm, hypotenuse = 5 cm → $\cos \theta = \frac{4}{5}=0.8$
$\tan \theta$	$\displaystyle \frac{\text{opposite}}{\text{adjacent}}$	If opposite = 3 cm, adjacent = 4 cm → $\tan \theta = \frac{3}{4}=0.75$

🔑 Remember: In a right triangle, the side opposite the right angle is the hypotenuse – it’s always the longest.

The Pythagorean Theorem – “$a^2 + b^2 = c^2$”

Think of the theorem as a recipe: add the squares of the two shorter sides, and you get the square of the longest side.

$$a^2 + b^2 = c^2$$

Example: A ladder 13 m long (hypotenuse) leans against a wall. The base is 5 m from the wall. Find the height.

Square the known sides: $5^2 = 25$, $13^2 = 169$
Subtract: $169 - 25 = 144$
Take the square root: $\sqrt{144} = 12$ m

Result: The wall is 12 m high.

Sine Rule – Solving Any Triangle

When you know two angles and one side, or two sides and a non‑included angle, the sine rule helps.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Example: In triangle ABC, $A=30^\circ$, $B=45^\circ$, side $a=8$ cm. Find side $b$.

Compute $C = 180^\circ - 30^\circ - 45^\circ = 105^\circ$
Use the rule: $\displaystyle \frac{8}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}$
$\sin 30^\circ = 0.5$, $\sin 45^\circ \approx 0.707$
$\displaystyle \frac{8}{0.5} = \frac{b}{0.707} \Rightarrow b \approx 11.3$ cm

Cosine Rule – When You Have Two Sides & Included Angle

Useful for any triangle when you know two sides and the angle between them.

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Example: In triangle ABC, $a=7$ cm, $b=9$ cm, angle $C=60^\circ$. Find side $c$.

Compute $\cos 60^\circ = 0.5$
$c^2 = 7^2 + 9^2 - 2(7)(9)(0.5)$
$c^2 = 49 + 81 - 63 = 67$
$c = \sqrt{67} \approx 8.2$ cm

Exam Tips 📚

Always check units – lengths in cm or m, angles in degrees.
Use a calculator for trigonometric values, but remember to set it to degree mode.
When applying the sine rule, ensure you use the correct angle opposite the side you’re solving for.
For the cosine rule, remember the minus sign before the $2ab\cos C$ term.
Show all steps – partial credit is awarded for correct methodology even if the final answer is slightly off.

Quick Flashcards – Keep These in Your Pocket

sin = opposite / hypotenuse

cos = adjacent / hypotenuse

tan = opposite / adjacent

Pythagoras = $a^2 + b^2 = c^2$

Sine Rule = $a/\sin A = b/\sin B = c/\sin C$

Cosine Rule = $c^2 = a^2 + b^2 - 2ab\cos C$

Trigonometry: sine, cosine, tangent, Pythagoras, sine and cosine rules