Law of Cosines and Sines MA 341 Brian

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Law of Cosines and Sines MA 341 Brian Oberg 30 NOV 99

Law of Cosines and Sines MA 341 Brian Oberg 30 NOV 99

Introduction • The objective here is to prove two points: • Law of Cosines:

Introduction • The objective here is to prove two points: • Law of Cosines: a^2 = b^2 + c^2 - 2 bc 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 =

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 • Given any triangle ABC • Label sides a, b, c

Law of Cosines • Draw in altitude from any angle to its base; this

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

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 -

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

Law of Cosines • h = b sin A • BD = c -

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

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 - 2 bc cos A + cos(A)^2

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

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

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

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

Law of Sines • sin A = h/b • h = b sin 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

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

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

Conclusion • Law of Cosines: a^2 = b^2 + c^2 - 2 bc cos

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