📘 GCSE Maths · Notes & worksheet
Trigonometry.
Notes
Right-angled triangles
SOH CAH TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
Non-right-angled triangles
Label sides with lowercase letters opposite the matching capital angle.
| Rule | Use when | Formula |
|---|---|---|
| Sine rule | you have a matching side-angle pair | a/sin A = b/sin B = c/sin C |
| Cosine rule (side) | two sides and the included angle | a² = b² + c² − 2bc cos A |
| Cosine rule (angle) | all three sides | cos A = (b² + c² − a²) / 2bc |
| Area | two sides and the included angle | Area = ½ab sin C |
For the sine rule, put the unknown on top.
Exact values (no calculator)
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
3D problems
- Find a right-angled triangle in the base and use Pythagoras to get a base diagonal.
- Make a second right-angled triangle that includes the height.
- The angle between a line and a plane uses the line and its projection on the plane.
Bearings
Measured clockwise from North, always with three figures (for example 077°). Draw North lines at each point and use "co-interior angles add to 180°" to relate forward and back bearings.
Common mistakes
- Calculator in radians instead of degrees. Always check the mode.
- Using the sine rule when you do not have a matching pair.
- Rounding too early. Keep full calculator values until the end.
- For the cosine rule, forgetting to square root at the end.
Exam tips
- Draw and label the triangle, even when a diagram is given.
- Ambiguous case: an angle found with the sine rule could be obtuse if it is opposite the larger side.
Worksheet
Prints the questions only, without answers. Choose “Save as PDF” in the print window to download.
Diagram placeholderThe diagrams for Q1–Q4, Q6 and Q7 are described in words in the questions. The drawn diagrams are being added to this page.
Total: 25 marks. Calculator allowed except Q5. Give answers to 3 significant figures unless stated.
- In triangle ABC, a = 8 cm, b = 11 cm and angle C = 40°. Find the length of side c. (3)
- In triangle ABC, angle A = 48°, angle B = 65° and a = 9.2 cm. Find the length of side b. (3)
- A triangle has two sides of length 7 cm and 9 cm with an included angle of 58°. Find its area. (2)
- A triangle has sides 5 cm, 7 cm and 9 cm. Find the size of the largest angle. (3)
- Without a calculator, show that tan 60° − tan 30° = 2/√3. (3)
- A cuboid ABCDEFGH has base ABCD with AB = 8 cm, BC = 6 cm and vertical height CG = 5 cm. Find the angle between the diagonal AG and the base ABCD. (4)
- A ship sails from port P for 12 km on a bearing of 040° to point Q. It then sails 9 km on a bearing of 130° to point R. Find the distance PR and the bearing of R from P. (5)
- Find the exact value of sin 60° × cos 30° + tan 45°. (2)
Show mark scheme and answers (tutor copy)
Try the worksheet first, then check your work here.
- c² = 8² + 11² − 2 × 8 × 11 × cos 40° (M1) = 185 − 134.82 = 50.18 (M1). c = 7.08 cm. (A1)
- b/sin 65° = 9.2/sin 48° (M1). b = 9.2 × sin 65° / sin 48° (M1) = 11.2 cm. (A1)
- Area = ½ × 7 × 9 × sin 58° (M1) = 26.7 cm². (A1)
- The largest angle is opposite the longest side (9 cm) (M1). cos θ = (25 + 49 − 81)/(2 × 5 × 7) = −7/70 = −0.1 (M1). θ = 95.7°. (A1)
- tan 60° = √3 and tan 30° = 1/√3 = √3/3 (M1). √3 − √3/3 = 2√3/3 (M1) = 2/√3 after multiplying top and bottom by √3 and simplifying (A1).
- Base diagonal AC = √(8² + 6²) = 10 cm (M1, A1). tan θ = 5/10 (M1). θ = 26.6°. (A1)
- The back bearing from Q to P is 220°. Angle PQR = 220° − 130° = 90° (M1). PR = √(12² + 9²) = 15 km (M1, A1). Angle QPR = tan⁻¹(9/12) = 36.9° (M1), so the bearing of R from P is 40° + 36.9° = 077° (A1).
- (√3/2)(√3/2) = 3/4, then 3/4 + 1 = 7/4. (M1, A1)