Pythagoras and trigonometry

📌 Key points

1. Pythagoras’ theorem: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). It only works in right-angled triangles.
2. SOH CAH TOA: sin θ = oppositehypotenuse, cos θ = adjacenthypotenuse, tan θ = oppositeadjacent.
3. To find an angle, use the inverse functions sin⁻¹, cos⁻¹ or tan⁻¹ on your calculator (in degree mode).
4. Bearings are measured clockwise from north and written with three figures, e.g. 045°. The back bearing differs by 180°.
5. Angles of elevation (looking up) and depression (looking down) are measured from the horizontal.
6. Sine rule: asin A = bsin B = csin C. Use it when you know a side and its opposite angle.
7. Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the included angle, or with all three sides to find an angle.
8. Area of any triangle = 12ab sin C, where C is the angle between sides a and b.
9. Exact values: sin 30° = 12, cos 60° = 12, tan 45° = 1, sin 60° = cos 30° = √32.
10. For 0° to 180°, sin x = sin(180° − x), so sin x = 0.5 has the two solutions 30° and 150°.

📖 Key terms

Hypotenuse

The longest side of a right-angled triangle, opposite the right angle.

Opposite side

The side across from the angle being used.

Adjacent side

The side next to the angle being used that is not the hypotenuse.

Bearing

An angle measured clockwise from north, written with three figures.

Angle of elevation

The angle measured upwards from the horizontal.

Angle of depression

The angle measured downwards from the horizontal.

Included angle

The angle between two given sides.