Geometry and Trigonometry
What this domain tests
This geometry and trigonometry chapter covers triangle congruence and similarity, circle properties, coordinate geometry, area and volume, and the trigonometric ratios extended to the unit circle. You should be able to prove relationships, apply the sine and cosine laws to non-right triangles, work in radians, and connect trigonometric functions to their graphs and identities.
Core rules and concepts
- Similarity: corresponding angles equal and sides proportional; areas scale as the square of the linear ratio, volumes as the cube.
- Circle facts: an inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of contact; the perpendicular from the centre bisects a chord.
- Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Sine law: a/sin A = b/sin B = c/sin C, with the ambiguous case (two possible triangles) when given two sides and a non-included angle.
- Cosine law: c² = a² + b² − 2ab cos C, used for SAS and SSS.
- Radians: π radians = 180°; arc length s = rθ and sector area A = ½r²θ for θ in radians.
- Identities: sin²θ + cos²θ = 1; tan θ = sin θ/cos θ.
Worked example
A triangle has sides a = 7, b = 9, and included angle C = 60°. Find side c.
Because we know two sides and the included angle (SAS), use the cosine law: c² = 7² + 9² − 2(7)(9)cos 60° c² = 49 + 81 − 126(0.5) c² = 130 − 63 = 67 c = √67 ≈ 8.19
Reasoning note: the minus sign in the cosine law is essential; using the sine law here would be invalid because no side–opposite-angle pair is known.
Traps that cost marks
- Using degree mode when the problem is in radians (or vice versa) in calculator work.
- Ignoring the ambiguous case of the sine law and reporting only one triangle.
- Confusing the linear scale factor with the area or volume factor in similarity problems.
- Mixing up inscribed angle and central angle relationships (central is double, not half).
- Applying arc length or sector formulas with θ in degrees instead of radians.