O-Level E-Math and SEC G3 Mathematics K310
G4: Pythagoras' theorem and trigonometry
Choose Pythagoras, sine, cosine, tangent, or a non-right-triangle method from the information given.
Core notes
Choose a triangle method from the information available. Pythagoras and basic trigonometric ratios require a right triangle; sine rule, cosine rule and the area formula extend the work to any triangle.
Pythagoras and right-triangle tests
For a right triangle, the square of the hypotenuse equals the sum of the squares of the shorter sides. Conversely, if the longest side satisfies that relationship, the triangle is right-angled.
Right-triangle trigonometry
Relative to the chosen acute angle, identify opposite, adjacent and hypotenuse. Use sine, cosine or tangent according to the sides involved, and use an inverse function to recover an angle.
Non-right triangles
Sine and cosine extend to obtuse angles. Use when two sides and their included angle are known. Use the sine rule with a known opposite pair, and the cosine rule for side-angle-side or three-side information.
Two-dimensional and three-dimensional problems
Draw a clear intermediate triangle for a solid, bearing or elevation problem. Bearings are measured clockwise from north using three figures. Angles of elevation and depression are measured from horizontal lines.
Formulae and methods
| Relationship | Use |
|---|---|
| Pythagoras' theorem with hypotenuse . | |
| Relate an acute angle to opposite and hypotenuse. | |
| Sine rule. | |
| Cosine rule. | |
| Area from two sides and the included angle. |
Worked examples
Example 1: In a triangle, , and . Find .
- The known angle is included between sides and , so use cosine rule.
- Write .
- Calculate , hence .
Answer: .
Example 2: A ladder of length reaches up a vertical wall. Find its distance from the wall.
- The ladder is the hypotenuse of a right triangle.
- Write .
- Calculate .
Answer: The foot is from the wall.
Chapter checkpoint
- Mark the right angle, known sides and known angles before selecting a method.
- Use the sine rule with opposite pairs and the cosine rule with the included angle or three sides.
- Check calculator angle mode and interpret bearings, elevation and depression from the correct reference line.
Exam traps and retrieval check
Avoid these traps
- Using right-triangle ratios in a triangle that is not right-angled.
- Mismatching a side with a non-opposite angle in the sine rule.
- Using degrees when the calculator is set to radians, or the reverse.
Check from memory
When is cosine rule a natural first choice?
With two sides and their included angle, or with all three sides.
From which direction is a bearing measured?
Clockwise from north.
What is the hypotenuse?
The side opposite the right angle and the longest side of a right triangle.
Official K310 coverage
This chapter maps to 7 official outcomes: G4.1, G4.2, G4.3, G4.4, G4.5, G4.6, G4.7. Check the official 2027 K310 syllabus for exact assessable wording and paper details.