Sine rule and cosine rule (Pure Maths) — A/L Combined Mathematics

What must I remember about Sine rule and cosine rule (Pure Maths)?

• Application of Sine and Cosine rules to heights, distances, and 3D surveying problems: angles of elevation (measured upward from horizontal) and angles of depression (measured downward from horizontal).
• Bearings in navigation: Three-figure bearing measured clockwise from North (000° to 360°); compass bearings specified as N θ° E, S θ° W, etc.
• Solving 3D geometrical configurations: Identify right-angled triangles and oblique triangles across horizontal and inclined vertical planes; use sine/cosine rules to express shared boundary segments.
• Circles associated with triangles: Circumcircle passing through all 3 vertices (center = circumcenter O at perpendicular bisectors' intersection); Incircle tangent to all 3 sides (center = incenter I at angle bisectors' intersection); Escribed circles (excircles) tangent to one side and extensions of the other two.
• Medians and Apollonius' Theorem: If AD is the median to side BC, then AB² + AC² = 2(AD² + BD²) = 2(AD² + (a/2)²).

Test yourself (1 minute)

1. Apollonius' Theorem proof via Cosine Rule

   Apply cosine rule to triangles ABD and ACD where cos(∠ADB) = -cos(∠ADC) because ∠ADB + ∠ADC = 180°; adding the two equations cancels the cross term to give AB² + AC² = 2(AD² + BD²).

2. Exradius formulas

   ra = Δ / (s - a) = s tan(A/2) = 4R sin(A/2) cos(B/2) cos(C/2); key identity: 1/ra + 1/rb + 1/rc = 1/r.

3. Three-dimensional angle determination

   When finding angle between a line and a plane, always drop a perpendicular from a point on the line to the plane, identifying the orthogonal projection.

Practise Sine rule and cosine rule (Pure Maths) questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 11

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Year 1 · Term 2