2Non right angled triangles What’s the relationship between a, b and C?Consider area S of right angled triangle, given by base and height
3The Sine RuleFor solving trianglesGivenCan show that…
4Solving triangles with sin rule Note all angles add up to 1800 (π radians)Note sin-1 can return θ and 180-θ hence possible ambiguityor =Though 132 invalid because >180
5Example 1 Two angles and any one side of a triangle are given In ABC of figure 4, A = 50, B = 70 and a = 10cm. Solve the triangle.(Answers correct to 3 significant figures三個有效數字 if necessary如必須.)Ca=10cm50o70oAB
6Solution From subtraction減法, C = 60o By sine formula, AC = 10 x sin 70o/ sin 50o = 12.3 cmAB = 10 x sin 60o/ sin 50o = 11.3 cm
7Two sides and one non-included angle are given Example 2 In ABC, find B if A = 30, b = 10cm and a = 4cm.
8sinB = 1.25 > 1 which is impossible不可能的 a=4cmb=10cmA=30oCsinB = 1.25 > 1 which is impossible不可能的Hence, no triangle exists for the data given. The situation 情況can further be illustrated 舉例證明 by accurate準確Bc
9Example 4In ABC, find B if A = 30, b = 10cm and a = 6cm.
10By sine formula, sinB = 10x sin30o/6 sinB = 0.8333 B = 56.4o, 123.6o SolutionBy sine formula, sinB = 10x sin30o/6sinB =B = 56.4o, 123.6oCb=10cma=6cmA=30oBB
11Cosine Rule By considering triangle can also show… When an angle and the two sides forming it are given use the cosine rule
12Given a triangle ABC, in which a = 28cm, c = 40cm, B = 35 Given a triangle ABC, in which a = 28cm, c = 40cm, B = 35. Find the length AC and correct your answer to 1 decimal place.
13Solution By cosine formula, b = 23.4 cm C b a=28cm 35o B A c=40cm Three sides are given35oBAc=40cm