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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 1800 (π radians) Note sin-1 can return θ and 180-θ hence possible ambiguity or = Though 132 invalid because >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.) C a=10cm 50o 70o A B

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**Solution From subtraction減法, C = 60o By sine formula,**

AC = 10 x sin 70o/ sin 50o = 12.3 cm AB = 10 x sin 60o/ sin 50o = 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 **

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

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

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**By sine formula, sinB = 10x sin30o/6 sinB = 0.8333 B = 56.4o, 123.6o**

Solution By sine formula, sinB = 10x sin30o/6 sinB = B = 56.4o, 123.6o C b=10cm a=6cm A=30o B B

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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**

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 C b a=28cm 35o B A c=40cm**

Three sides are given 35o B A c=40cm

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