Geometry and Trigonometry
Geometry and Trigonometry
Geometry and Trigonometry tests angle reasoning, polygons, circles, congruence and similarity, Pythagoras’ theorem, right-triangle trigonometry, the sine and cosine rules, and bearings.
Core rules
- Angles on a straight line sum to 180°, angles at a point to 360°, and vertically opposite angles are equal.
- The interior angle sum of an n-sided polygon is (n − 2) × 180°; for a regular polygon each exterior angle is 360°/n.
- Pythagoras: a² + b² = c² for the hypotenuse c. SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- For non-right triangles, use a/sin A = b/sin B = c/sin C and c² = a² + b² − 2ab cos C.
- Bearings are measured clockwise from north and written as three digits.
Worked example
A triangle has sides 7 cm and 9 cm with included angle 60°. The third side c satisfies c² = 7² + 9² − 2(7)(9)cos 60° = 49 + 81 − 63 = 67, so c = √67 ≈ 8.19 cm. The cosine rule is needed because the angle is between two known sides, not opposite a known side.
Traps
- Using the sine rule when the given angle is included between two sides; use the cosine rule.
- The ambiguous case of the sine rule can give two possible angles.
- Mixing up opposite and adjacent sides after rotating a diagram.
- Measuring a bearing anticlockwise or omitting the leading zero.