This module covers triangles that have no right angle: the sine rule, the cosine rule, the area formula ½ab sin C, and bearings with journeys made of two legs. It also covers the ambiguous case, where the same measurements can give two different triangles.
Check the current Cambridge IGCSE Mathematics 0580 syllabus for which of these points belong to your tier. Bearings are usually met earlier than the sine and cosine rules, so confirm your own route there. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need right-triangle trigonometry: sin, cos and tan, Pythagoras, and using the calculator in degree mode. You also need angle facts for parallel lines and angles in a triangle summing to 180°. If those feel shaky, return to them first, because bearings rely on parallel north lines.
An orienting example
A boat sails 10 km on a bearing of 040°, then 14 km on a bearing of 120°. How far is it from the start?
Step 1, draw: sketch the start, the turning point and the finish, with a north line at each point where a bearing begins.
Step 2, find the angle inside the triangle: at the turning point, the back bearing to the start is 220°, and the second leg is at 120°. The angle between them is 220° − 120° = 100°.
Step 3, choose a rule: two sides and the included angle, so the cosine rule. d² = 10² + 14² − 2 × 10 × 14 × cos 100° = 344.6, so d = 18.6 km.
That one question used bearings, parallel lines, a rule choice and a calculator step. It is a good picture of how the module hangs together.
In which order should you study it?
- Choose a sine or cosine relationship: the decision that every later lesson relies on.
- Use bearings with north lines consistently: the diagram habit that stops angle errors.
- Resolve a two-stage journey diagram: puts the first two lessons together in one triangle.
- Find an area from two sides and an included angle: a short, useful formula that reuses the same sine values.
- Explain an ambiguous diagram before calculating: the case where one calculator answer is not enough.
Then work through the mixed practice set. A steady pace is one lesson a day, with the practice set at the weekend.
Which traps catch most students here?
- Picking the rule from the question instead of from the given information.
- Subtracting before multiplying in the cosine rule.
- Measuring a bearing anticlockwise, or leaving off the zero at the front.
- Using the wrong interior angle at a turning point.
- Using an angle that is not between the two sides in ½ab sin C.
- Stopping at the acute angle when the diagram allows an obtuse one.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Sketch every diagram on paper before opening an answer, and write the rule you are using in words. The triangle and bearings reasoning board asks for the same habit. When you get a question wrong, use the routing notes at the end of the practice set and go back to the named lesson.
Fix the lesson, then retry a fresh question a few days later. If your next topic is vectors, the vectors and transformations module follows, and our online one-to-one Mathematics tuition is there if you want a teacher to look at your written working.