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Contents
Cosine Rule
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Cosine Rule Case 1 (If A is an acute angle) In
the figure shown, consider angle A. a2 = h2 + (c - x)2
= h2 + c2 - 2cx + x2
b2 = h2 + x2
a2 = b2 + c2 - 2cx
Also cos
A = x/b
x = b cos A
a2 = b2 + c2 - 2bc cos
A
Case 2
(If A is an obtuse angle)
 The
figure on the right show ÐBAC as obtuse.
a2 = h2 + (c + x)2
= h2 + c2 + 2cx + x2
b2 =
h2 + x2
a2 =
b2 + c2
+ 2cx
Also cos A = x/b
x = b cos ÐCAN = b cos (180° -
A) = -b cos A
a2 =
b2 + c2 - 2bc cos A
Hence in either triangle, a2 = b2 +
c2
- 2bc cos A
This is the cosine rule for any angle A. By taking angles B and
C,
two similar formulae can be derived the three formulae are:
a2 = b2 + c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C
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