Coordinate Geometry and Trigonometry
This domain combines straight-line and circle geometry with radian measure, trigonometric identities, trigonometric equations and triangle solution. CAPE rewards candidates who choose the most efficient identity and who track the required solution interval carefully.
Lines and circles
Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal gradients, perpendicular lines satisfy m₁m₂ = −1. The circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². Expanding x² + y² + 2gx + 2fy + c = 0 gives centre (−g, −f) and radius √(g² + f² − c). A tangent is perpendicular to the radius at the point of contact.
Radians, arcs and sectors
With θ in radians: arc length s = rθ, sector area A = ½r²θ, and triangle area ½r² sin θ. Convert using π radians = 180°.
Identities and equations
sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; sec²θ = 1 + tan²θ; cosec²θ = 1 + cot²θ. sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B). Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1.
For a cos θ + b sin θ, write R cos(θ − α) with R = √(a² + b²) and tan α = b/a.
Triangle rules
Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A. Area = ½ab sin C.
Worked example
Solve 3 sin θ + 4 cos θ = 2 for 0° ≤ θ ≤ 360°. Write 3 sin θ + 4 cos θ = 5 cos(θ − α), where R = √(3² + 4²) = 5, cos α = 4/5 and sin α = 3/5, so α ≈ 36.87°. Then cos(θ − 36.87°) = 0.4, giving θ − 36.87° = ±66.42° + 360°n. So θ ≈ 103.3° or θ ≈ 330.5°, both inside the interval.
Traps
- Calculator left in the wrong angle mode.
- Dividing by cos θ and losing solutions.
- Giving only the principal value when the interval admits more.
- Forgetting the negative root when solving cos θ = k or sin θ = k.