Solution of triangles asks you to find missing sides, angles and areas in triangles that are not right-angled. This chapter of SPM Additional Mathematics has four linked skills, and each one starts with reading what the question gives you.
What does this chapter cover?
The skills build on each other. Choosing the right rule comes first, because the ambiguous case, area and 3D problems all begin with that choice.
| Skill | Question it answers |
|---|---|
| Choosing sine rule or cosine rule | Which rule can start from the given sides and angles? |
| The ambiguous sine-rule case | Can the same data make zero, one or two triangles? |
| Area with a non-included height | How do I find area when the given angle is not between the given sides? |
| Three-dimensional applications | Which flat triangle inside the diagram do I solve first? |
How do the skills connect?
One small example shows the link. Triangle ABC has AB = 7 cm, AC = 9 cm and angle A = 60°, and you want BC.
The known angle sits between the two known sides, so the cosine rule starts at once: BC² = 7² + 9² − 2(7)(9)cos 60° = 130 − 63 = 67, so BC ≈ 8.19 cm. The sine rule cannot start here, because no side and its opposite angle are both known.
Change the question to “AB = 7 cm, angle A = 60°, angle C = 50°, find BC” and the sine rule takes over. The same triangle type needs a different rule once the given information changes.
Where should I start?
Pick the row that matches your situation.
- Both formulas are memorised but you hesitate at the start: begin with choosing the rule.
- Answers sometimes come out as an impossible sine value: read the ambiguous case.
- The angle you are given is not between the two sides: try the area lesson.
- Diagrams with towers, poles or pyramids feel crowded: finish with the 3D lesson.
When you have read the lessons, test the whole chapter with the solution of triangles practice set. If you want a teacher to watch your method on mixed questions, see online one-to-one Additional Mathematics tuition.