Form 4 Additional Mathematics Bab 9: Solution of Triangles
9.1 Sine Rule
Sine rule: a / sin A = b / sin B = c / sin C (or sin A / a = sin B / b = sin C / c). (p.243)
Use the sine rule when given two angles and one side, or two sides and a non-included angle. (p.245)
Bearing is the direction of a place from a reference point, measured clockwise from north. (p.245)
Ambiguous case: two different triangles can be formed from two sides and a non-included angle. (p.246)
Ambiguous case exists if the non-included angle is acute and the side opposite it (a) satisfies h < a < c, where h is the height of the triangle. If a < h no triangle exists; if a = h or a ≥ c one triangle exists. (p.247)
In the ambiguous case the two possible angles are θ and 180° − θ. (p.248)
9.2 Cosine Rule
Use the cosine rule when given two sides and the included angle, or three sides. (p.251)
a2 = b2 + c2 − 2bc cos A; b2 = a2 + c2 − 2ac cos B; c2 = a2 + b2 − 2ab cos C. (p.251)
To find angles: cos A = (b2 + c2 − a2) / 2bc, and similarly for B and C. (p.253)
9.3 Area of a Triangle
Area = ½ab sin C = ½ac sin B = ½bc sin A (two sides and the included angle). (p.257)
Heron's formula: Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2. (p.259)
9.4 Applications
Choose the rule by the information given: sine rule (angle-side pairs), cosine rule (SAS or SSS), area formulae or Heron's formula for area. (p.263)
In three-dimensional problems, identify the right triangles and the angle between a line and a plane before applying the rules. (p.264)