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Sine and cosine formula

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Non right angled triangles What’s the relationship between a, b and C? Consider area S of right angled triangle, given by base and height

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The Sine Rule For solving triangles Given Can show that…

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Solving triangles with sin rule Note all angles add up to 180 0 (π radians) Note sin -1 can return θ and 180-θ hence possible ambiguity 47.610 0 or 180-47.61=132.390 0 Though 132 invalid because 132+80>180

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Example 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.) 50 o 70 o a=10cm AB C

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Solution From subtraction 減法, C = 60 o By sine formula, AC = 10 x sin 70 o / sin 50 o = 12.3 cm AB = 10 x sin 60 o / sin 50 o = 11.3 cm

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Two sides and one non-included angle are given Example 2 In ABC, find B if A = 30 , b = 10cm and a = 4cm.

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sinB = 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 a=4cm b=10cm A=30 o C c B

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Example 4 In ABC, find B if A = 30 , b = 10cm and a = 6cm.

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Solution By sine formula, sinB = 10x sin30 o /6 sinB = 0.8333 B = 56.4 o, 123.6 o b=10cm A=30 o a=6cm BB C

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Cosine Rule By considering triangle can also show… When an angle and the two sides forming it are given use the cosine rule

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Given a triangle ABC, in which a = 28cm, c = 40cm, B = 35 . Find the length AC and correct your answer to 1 decimal place.

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Solution By cosine formula, b = 23.4 cm a=28cm 35 o c=40cm C A B b

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