# Sine and cosine formula

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Sine and cosine formula

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

The Sine Rule For solving triangles Given Can show that…

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

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

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

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

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

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

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

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

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.

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