Question

Using Cramer's rule...

Answer 1

take a look the picture below. Help me for number 5. a & b

Answer 2

have you idea, guys?

Answer 3

Cramer's rule states that for a system of equations \[Ax = b\]the soutions is going to be \[x_i = \frac{ \det(A_i) }{ \det(A) }\]where A_i is A with column i replaced by b.

First you have construct matrix A and vectors x and b.
Since x_i = cos(alpha) the x-vector has to contain the cosines, and from this you get A as well. b is given as the answers to the equations.

We know \[0 \cos(\alpha) + c \cos(\beta) + b \cos(\gamma) = a\]\[c \cos(\alpha) + 0 \cos(\beta) + a \cos(\gamma) = b\]\[b \cos(\alpha) + a \cos(\beta) + 0 \cos(\gamma) = c\]
From this we get \[x = \left(\begin{matrix}\cos(\alpha) \\ \cos(\beta) \\ \cos(\gamma) \end{matrix}\right)\]\[b = \left(\begin{matrix}a \\ b \\ c\end{matrix}\right)\]\[A =

\left[\begin{matrix}0 & c & b \\ c & 0 & a \\ b & a & 0 \end{matrix}\right]\]A_i is A with i:th column replaced by b. Since we want\[x_i = \cos(\alpha) \rightarrow i = 1\]ee get\[A_i = A_1 = \left[\begin{matrix}a & c & b \\ b & 0 & a \\ c & a & 0 \end{matrix}\right]\]
From this you can caluculate\[x_i = \frac{ \det(A_i) }{ \det(A) } \rightarrow x_1 = \cos(\alpha) = \frac{ \det(A_1) }{ \det(A) }\]

Answer 4

For cos(beta) and cos(gamma) you just change A_i and revaluate the determinat.

Answer 5

i got it.., thank you @Lyrae :)

Answer 6

Yw! anytime :)