3.2 Law Of Cosines
If A, B and C are the measure of angles of a triangle
ABC and a, b and c are lengths of sides opposite the angles A, B
and C respectively, then
These three formulas are called the Law of cosines
or cosine rule. Let us place angle A of triangle ABC in standard
position. Further Ð A < 180^{0}
therefore point C is above xaxis. In two figures, Ð
A is as acute and obtuse.
^{.}.^{. } AB = C \
B º (c, 0). Let C º
(x, y). Since AC = b.
We have x = b cos A and y = b sin A
\ C º
( b cos A, b sin A). Now BC = a
By using distance formula
(BC)^{2 } = (c  b cos A)^{2 } + (b sin A)^{2
}
\ a^{2 } = c^{2 } 
2bc cos A + b^{2 } cos^{2 } A + b^{2 } sin^{2
} A
\ a^{2 } = c^{2 } 
2bc cos A + b^{2 } (cos^{2 } A + sin^{2 }
A)
\ a^{2 } = c2  2bc cos A +
b^{2 } (1)
\ a^{2 } = c^{2 } 
2bc cos A + b^{2 }
\ a^{2 } = b^{2 } +
c^{2 }  2bc cos A OR
\
Note that if A = 90^{0 }, then cos A =
cos 90^{0} = 0, giving a^{2} = b^{2} + c^{2}
which is the "Pythagoras theorem" for right triangles.
If the orientation of the triangle is changed to have B or C at
the origin, then the other two versions of Law of cosines can be
obtained.
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