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This article expands the previously written article, Understanding the Pythagorean theorem. It discusses the derivation the law of cosines, which is derived by the Pythagorean identity.
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Many of us may have come across this formula since secondary school, but has anyone been curious about the derivation of this formula? This formula allows us to find dimensions with just one angle and two sides.

Since many real life triangles are not right-angled, this general rule has been investigated by many mathematicians. Euclid (fl. 300BC), a Greek geometer, is one of the pioneers for the law of cosines. One of his publications, Euclid's Elements, contains the earlier version of the law of cosines.
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Application

Length of a can be found with two other sides and an opposite angle with respect to line BC.
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Geometry

Right-angled triangle, BCD
Obtuse line to form obtuse triangle, AB
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Lengths

BC = a
AC = b
AB = c
AD = d
BD = e
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Derivation

Basic trigonometrical identities needed are as such,
- cos ( alpha) = cos (alpha)
sin ( alpha) = sin (alpha)

From the diagram, we see that
d = -c cos ()
e = c sin ()

By the Pythagorean identity,
a^2 = (b + d)^2 + e^2
a^2 = (b^2 + 2bd + d^2) + e^2
a^2 = b^2 + 2b[-c cos (alpha)] + [-c cos(alpha)] ^2 + [c sin (alpha)] ^2
a^2 = b^2 - 2bc cos (alpha) + c^2 cos( alpha) ^2 + c^2 sin^2 (alpha)
a^2 = b^2 + c^2 [sin^2 (alpha) + cos^2 (alpha)] 2bc cos(alpha)

Which gives,
a^2 = b^2 + c^2 - 2bc cos(alpha) > the Law of Cosines (Cosine Rule)
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Learn more about this author, Freddy Tay.
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