Thread: Help to verify & solve this determinants problem!

Originally Posted by test2k6

Verify the system of linear equations in cosA, cosB, and cosC.

_______ c cosB + b cosC = a
c cosA ______ + a cosC = b
b cosA + a cosB _______ = c

Then use Cramer's Rule to solve for cosC, and use the result to verify the Law of Cosines, c^2 = a^2 + b^2 - 2ab cosC

c cosB + b cosC = a
c cosA + a cosC = b
b cosA + a cosB = c

write this in the form: Ax = b, where A, x, b are matrices
so we have:

|0......c......b| |cosA|.......|a|
|c......0......a| |cosB|...=...|b|
|b......a......0| |cosC|.......|c|

Now detA = 0 + abc + abc - 0 - 0 - 0 = 2abc

By Cramer's rule:

cosC = det |0......c......a| divided by detA = 2abc
................|c......0......b|
................|b......a......c|

=> cosC = (0 + cb^2 + ca^2 - 0 - 0 - c^3)/2abc
=> cosC = (cb^2 + ca^2 - c^3)/2abc
=> cosC = (b^2 + a^2 - c^2)/2ab ..............factored out and canceled the c's
=> 2abcosC = a^2 + b^2 - c^2
=> c^2 = a^2 + b^2 - 2abcosC which is the cosine rule

Originally Posted by test2k6

Verify the system of linear equations in cosA, cosB, and cosC.

c cosB + b cosC = a
c cosA + a cosC = b
b cosA + a cosB = c

I must admit I'm curious as to where these equations come from, myself.

-Dan