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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