Law of Cosines and Sines MA341 Brian Oberg 30 NOV 99

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

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

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

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

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

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

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

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

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^2 - 2bc cos A

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

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

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

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

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