An elementary proof of the cosine law
The cosine law states that in triangle with side lengths
and
, the following equations are satisfied:
*
The discussion below shows how these equations were derived.
***
Consider triangle with side lengths
, and
. Drop an altitude
from
and let
be the intersection of
and
as shown in the figure below. If we let
be the length of
, then
.
(1) Remembering the mnemonic SOHCAHTOA, in a triangle, the cosine of an angle is equal to length of the side adjacent to it divided by the length of the hypotenuse. Therefore, in triangle ,
. Simplifying, we have
.
(2) In triangle , by the Pythagorean Theorem,
which when simplified equals to
.
(3) Also, in triangle , by the Pythagorean theorem,
.
(4) Rearranging the equation in step (2), we have . Now, substituting the equation in step (3) to the preceding equation and rearranging the terms, we have
.
(5) Substituting the equation in step (1) to the equation in step (4), we have
, which is equal to the first equation above *.
The formula for and
follow from the proof above. This proves the cosine law.
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