Pythagoras' Theorem and Trigonometry
Pythagoras' Theorem
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:
$$c^2 = a^2 + b^2$$
where $c$ is the hypotenuse (the side opposite the right angle — always the longest side).
Finding the hypotenuse: $$c = \sqrt{a^2 + b^2}$$
Finding a shorter side: $$a = \sqrt{c^2 - b^2}$$
Example: A right-angled triangle has legs of 5 cm and 12 cm. Find the hypotenuse. $$h = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}$$
Pythagorean triples — integer solutions to $a^2 + b^2 = c^2$:
- 3, 4, 5 (and multiples: 6, 8, 10; 9, 12, 15)
- 5, 12, 13
- 8, 15, 17
Trigonometric Ratios (SOHCAHTOA)
In a right-angled triangle, for angle $\theta$:
$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}$$ $$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}$$ $$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$$
Memory aid: SOH CAH TOA
Finding a side:
$$\text{side} = \text{hypotenuse} \times \sin\theta \quad \text{(to find opposite)}$$ $$\text{side} = \text{hypotenuse} \times \cos\theta \quad \text{(to find adjacent)}$$
Example: Find side $x$ in a right triangle with hypotenuse 10 cm and angle 35°. $$x = 10 \sin 35° = 10 \times 0.5736 \approx 5.74 \text{ cm}$$
Finding an angle:
Use the inverse trig function: $$\theta = \sin^{-1}\left(\frac{O}{H}\right), \quad \theta = \cos^{-1}\left(\frac{A}{H}\right), \quad \theta = \tan^{-1}\left(\frac{O}{A}\right)$$
Example: In a right triangle, opposite = 7 and adjacent = 10. Find the angle. $$\theta = \tan^{-1}\left(\frac{7}{10}\right) = \tan^{-1}(0.7) \approx 35.0°$$
Exact Values (Higher Tier)
| Angle ($\theta$) | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| 45° | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | 1 |
| 60° | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| 90° | 1 | 0 | undefined |
Angles of Elevation and Depression
- Angle of elevation: angle measured upward from horizontal
- Angle of depression: angle measured downward from horizontal
Both are measured from the horizontal line of sight.
WJEC Exam Tips
- Pythagoras: always identify the hypotenuse first — it is opposite the right angle.
- SOHCAHTOA: label sides O, A, H relative to the angle given.
- Calculator: ensure it is in degree mode for these problems.
- Show: state which ratio you are using before substituting.
AI-generated · claude-opus-4-7 · v3-wjec-maths