Geometry and Trigonometry
Geometry and Trigonometry covers right-angled and non-right-angled triangles, the trigonometric ratios, the sine and cosine rules, area, and radian measure for arcs and sectors. Diagrams are often not to scale, so use the given measurements.
Right triangles
- Pythagoras: a² + b² = c², where c is the hypotenuse.
- SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Use inverse trig functions to find an angle.
Non-right triangles
- Sine rule: a/sin A = b/sin B = c/sin C.
- Cosine rule: a² = b² + c² − 2bc cos A; rearrange to find an angle.
- Area = ½ab sin C, where C is the included angle.
Radians, arcs and sectors
π radians = 180°. Arc length s = rθ and sector area A = ½r²θ, with θ in radians.
Worked example
A triangle has sides 7 cm and 8 cm with an included angle of 60°. Area = ½ × 7 × 8 × sin 60° = 28 × 0.8660 = 24.25 cm² (to 2 d.p.). The third side c satisfies c² = 7² + 8² − 2(7)(8)cos 60° = 49 + 64 − 56 = 57, so c = √57 = 7.55 cm.
Traps
- Calculator in the wrong mode (degrees versus radians).
- Using the sine rule when the cosine rule is needed (two sides and the included angle).
- Using a side that is not included between the two sides in the area formula.
- Rounding too early and losing accuracy.