Geometry and Trigonometry
What the exam tests
Plane geometry and trigonometry: triangle similarity and congruence, circle theorems, areas, the sine and cosine rules, trigonometric identities and equations, radian measure, and simple coordinate geometry. Questions often combine a geometric configuration with a trigonometric computation.
Core rules
- The angles of a triangle sum to 180°; an exterior angle equals the sum of the two opposite interior angles.
- Sine rule: a/sin A = b/sin B = c/sin C = 2R.
- Cosine rule: c²=a²+b²−2ab cos C. Use it when you know two sides and the included angle, or all three sides.
- Area = ½ab sin C.
- Inscribed angle theorem: an angle subtended by a diameter is 90°; a tangent is perpendicular to the radius at the point of contact.
- General solutions: sin x=sin α gives x=α+2kπ or x=π−α+2kπ; cos x=cos α gives x=±α+2kπ.
Worked example
In triangle ABC, a=7, b=8 and C=60°. Find c and the area. c²=7²+8²−2(7)(8)cos 60°=49+64−56=57, so c=√57≈7.55. Area=½(7)(8)sin 60°=28·(√3/2)=14√3≈24.2 square units.
Traps
- Using the sine rule in an SSA configuration and missing the obtuse-angle solution.
- Leaving the calculator in degree mode for a radian problem, or the reverse.
- Forgetting the minus sign in the cosine rule.
- Losing solutions by dividing a trigonometric equation by sin x or cos x.
- Assuming a triangle is right-angled because a diagram looks so.