1. Law of Cosines and Sines
MA341
Brian Oberg
30 NOV 99

2. Introduction The objective here is to prove two points:
Law of Cosines: a^2 = b^2 + c^2 - 2bc cos A
Law of Sines: a / sin A = b / sin B = c / sin C

3. Given Terms sin(A)^2+cos(A)^2 =1 sin A = a/c cos A = b/c
Pythagorean relation: a^2+b^2=c^2

4. Law of Cosines
Given any triangle ABC
Label sides a,b,c

5. Law of Cosines
Draw in altitude from any angle to its base; this forms two right triangles
Label altitude h and the new point D

6. Law of Cosines
sin A = h/b
h = b sin A
cos A = AD/b
AD = b cos A

7. Law of Cosines
AD = b cos A
BD = c - AD
BD = c - b cos A

8. Law of Cosines h = b sin A BD = c - b cos A a^2 = h^2 + (BD)^2
a^2 = (b sin A)^2 + (c-b cos A)^2

9. Law of Cosines a^2 = (b sin A)^2 + (c - b cos A)^2
(b sin A)^2 = b^2 sin(A)^2
(c - b cos A)^2 = c^2 - 2bc cos A cos(A)^2

10. Law of Cosines a^2 = b^2 sin(A)^2 + c^2 - 2bc cos A + cos(A)^2
a^2= b^2(sin(A)^2 + cos(A)^2) + c^ bc cos A

11. Law of Cosines a^2= b^2(sin(A)^2 + cos(A)^2) + c^2 - 2bc cos A
remember: sin(A)^2 + cos(A)^2=1
a^2 = b^2 + c^2 - 2bc cos A

12. Law of Sines
sin A = h/b
h = b sin A
sin B = h/a
h= a sin B

13. Law of Sines h = b sin A h= a sin B a sin B = b sin A
a / sin A = b / sin B

14. Law of Sines
Furthermore: a / sin A = b / sin B = c / sin C

15. Conclusion Law of Cosines: a^2 = b^2 + c^2 - 2bc cos A
Law of Sines: a / sin A = b / sin B = c / sin C