I have been teaching Precalculus/Trigonometry for 18 years and I just figured out a much simpler way to derive the Law of Cosines that the students can actually do themselves, with a little guidance from me perhaps, instead of watching me do a more complicated version while their eyes glaze over and groans ensue.
Here it is:
Start with a right triangle and recall the Pythagorean Theorem. Then draw a second triangle where sides a and b are the same, but side c is shorter because angle C is less than 90 degrees. We don’t know how (yet) to find side C, so we divide triangle 2 into two right triangles and then we can use the Pythagorean Theorem to write a system of equations. Solving the system of equations gives us c in terms of a, b, and x. Then find x using cosine of the known angle C (let students know up front that the given information is the measure of angle C and the side lengths a and b (a SAS triangle).
I used the same diagram in previous years, but I didn’t use a system of equations to simplify the relationship first, so the substitutions used both sine and cosine and required multiplying out binomials with sine and cosine in them and simplifying with the Pythagorean trig property. The derivation used a lot of nice math, but was so long and cumbersome that it became meaningless to the kids and the first question out of their mouths was “Do we have to know how to do this for the test?” rather than, “Oh, that makes sense,” which is the response that I hope this newer method will elicit.
ANOTHER EPIPHANY! I have students who forget they can use the Law of Cosines and who would divide triangle 2 into two right triangles to find side C anyway. So I should let them find side C the long way and then ASK THEM IF THEY WOULD LIKE TO KNOW A QUICKER WAY TO DO THIS! The old Dan Meyer aspirin for a headache approach.
After deriving the formula: Then follow up by actually constructing 3 triangles with the same sides a and b but different angles for C and show how the term abcosC is the amount of change (+ or -) in the length of side C and that, when C = 90 degrees, abcosC = 0, so the Law of Cosines is just a special case of the Pythagorean Theorem.