Proof of law of cosines
The possible cases are
(1) is acute (all angles are acute)
(2) is obtuse (one obtuse angle)
(3) is right
Within cases (2) and (3), the choice of which angle is obtuse or right is arbitrary; it's not going to make difference if , , or as the triangle can be oriented and labeled any way.
Case 1: is acute ()
1. Orient the triangle any way you like (I'll put vertex at the top), and construct an altitude, dividing side length into .
2. Use the Pythagorean Theorem to relate .
[Fill in 1]
3. Use the Pythagorean Theorem to relate , and solve for in terms of the other two sides.
[Fill in 2]
4. Substitute, so you have an expression for in terms of .
[Fill in 3]
5. Add and subtract a term on the right hand side.
[Fill in 4]
6. And observe that . Make a substitution into the subtracted . You should now have an expression
[Fill in 5]
7. Which, if things have been going well up to this point, should simplify to
[Fill in 6 - Show algebra]
(If it doesn't, you may need to go back and reconsider what you've got above.)
8. By definition, [Fill in 7], and so [Fill in 8]. Also, .
9. Substitute, and end up with the law of cosines.
[Do that.]
Since all angles acute, the proof for this case holds no matter how the triangle is oriented, and we also get
Case 2: is obtuse (say)
Suppose is obtuse, and we'll arbitrarily say that is the obtuse angle.
As before, with the vertex at the top, the proof of the
variant is identical to Case 1, and the proof of the
is the same (it's the mirror image on the other side). The one we need to establish is the
for ,
Orient the triangle so it sits on base , and construct an altitude from vertex . This requires you to extend as shown.
2. Use the Pythagorean Theorem to relate . Solve for .
[Fill in 1]
3. Use the Pythagorean Theorem to relate .
[Fill in 2]
4. Substitute, so you have an expression for in terms of .
[Fill in 3]
5. Simplify. The terms should cancel.
[Fill in 4]
6. Since is obtuse, we define its cosine in terms of its supplement; by definition [Fill in 5].
7. Since = [Fill in 6], [Fill in 7], and [Fill in 8].
8. Substitute, and end up with the law of cosines.
[Do that.]
The law of cosines is now established for this case.
Case 3: is right (say)
as in Case 1.
1. With , the Pythagorean Theorem gives us (using lengths marked on triangle):
[FIll in 1]
2. Also, since , by definition, [Fill in 2]
3. So the term is equal to [Fill in 3] as well, so we could add it in, and get the law.
[Do that.]
And...done.