GCSE Further Maths · Topic 5
Geometry and trigonometry
Trigonometry works for any triangle, not just right-angled ones.
We will use the sine and cosine rules and the area formula, including in 3D.
Then we meet trigonometric graphs, identities and equations, and measure angles in radians.
- 3 subtopics
- 138 questions
- Free: 5.1 notes and 10 questions
In this topic
- 5.1The sine and cosine rules, area of a triangle and 3D trigonometryFree below · 52 questions
- 5.2Trigonometric graphs, identities and equationsOpen in Full Marks · 35 questions
- 5.3Circular measure: radians, arc length and sector areaOpen in Full Marks · 51 questions
5.1 · Free notes
The sine and cosine rules, area of a triangle and 3D trigonometry
Key points
- Sine rule: a/sin A = b/sin B = c/sin C. Use it when you know a side and its opposite angle.
- Cosine rule: a² = b² + c² − 2bc cos A; rearranged, cos A = (b² + c² − a²)/(2bc). Use it with two sides and the included angle, or all three sides.
- Area of a triangle = ½ab sin C, where C is the angle between sides a and b.
- The sine rule can give two possible angles (the ambiguous case): if sin B = 0.6, B could be 36.9° or 143.1°.
- In 3D problems, find a right-angled or other triangle in the shape, and use Pythagoras and trigonometry in 2D steps; the angle between a line and a plane is found from the line's projection onto the plane.
In more detail
- Worked example (3D): a cuboid 3 by 4 by 12. The base diagonal is 5; the angle between the space diagonal (13) and the base is tan⁻¹(12/5) ≈ 67.4°.
- Ambiguous case check: the second angle 180° − B is only possible if it leaves room for the other given angle (the three angles must add to 180°).
- Exam tip: label the triangle with a opposite A, b opposite B and c opposite C before substituting.
Practice
The sine and cosine rules, area of a triangle and 3D trigonometry: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. In triangle ABC, a = 8, A = 30°, B = 45°. Which rule finds b?
- Cosine rule
- Pythagoras
- Area rule
- Sine rule
Show answer
D: Sine rule. A side with its opposite angle is known.
2. Find the area of a triangle with sides 6 and 10 and included angle 30°.
- 30
- 15
- 60
- 7.5
Show answer
B: 15. ½ × 6 × 10 × sin 30° = 15.
3. In a triangle, b = 5, c = 7, A = 60°. Find a² using the cosine rule.
- 39
- 74
- 109
- 35
Show answer
A: 39. 25 + 49 − 70 × 0.5 = 39.
4. Which formula gives an angle from three sides?
- sin A = a/b
- a/sin A = b/sin B
- cos A = (b² + c² − a²)/(2bc)
- ½ab sin C
Show answer
C: cos A = (b² + c² − a²)/(2bc). The rearranged cosine rule.
5. If sin B = 0.5 (B in a triangle), what are the possible values of B?
- 30° only
- 60° or 120°
- 45° or 135°
- 30° or 150°
Show answer
D: 30° or 150°. sin(180° − B) = sin B.
6. A triangle has sides 3, 5 and 7. What is the largest angle?
- 90°
- 120°
- 60°
- 135°
Show answer
B: 120°. cos C = (9 + 25 − 49)/30 = −0.5.
7. Using the sine rule, a = 10, A = 30°, B = 90°. What is b?
- 20
- 5
- 10
- 17.3
Show answer
A: 20. b = 10 sin 90°/sin 30° = 20.
8. What is the area of an equilateral triangle with side 4?
- 8
- 16
- 4√3
- 2√3
Show answer
C: 4√3. ½ × 4 × 4 × sin 60° = 8 × √3/2.
9. In a cuboid 3 × 4 × 12, what is the length of the space diagonal?
- 19
- 12
- 5
- 13
Show answer
D: 13. √(9 + 16 + 144) = 13.
10. When should you use the cosine rule rather than the sine rule?
- A side and its opposite angle are known
- Two sides and the included angle are known
- Only right angles are involved
- Never
Show answer
B: Two sides and the included angle are known. The included angle case needs the cosine rule.
Keep going in Full Marks
Full Marks has 52 questions on The sine and cosine rules, area of a triangle and 3D trigonometry (42 more than here), set out in practice levels. You also get notes and 3D models for all 3 subtopics of Geometry and trigonometry, plus games, timed exams and past papers.