Question

follow these steps to derive the law of cosines.

1. in $\triangle abd$, the trigonometric function $\cos(a)=\frac{x}{c}$.
2. multiply both sides of the equation in step 5 by $c$ to get $x = c \cos(a)$.
3. substitute $c \cos(a)$ for the variable $x$ in the equation $a^2 = b^2 - 2bx + c^2$ to produce $a^2 = b^2 - 2bc \cos(a) + c^2$.
4. the law of cosines is obtained by ______ the previous equation.

$a^2 = b^2 + c^2 - 2bc \cos(a)$

Explanation:

Step1: Identify target equation form

We need to rearrange $a^2 = b^2 - 2bc\cos(A) + c^2$ to match $a^2 = b^2 + c^2 - 2bc\cos(A)$.

Step2: Rearrange terms by commutative property

The commutative property of addition states $x + y = y + x$. Apply this to the terms $b^2$ and $c^2$:
$a^2 = b^2 + c^2 - 2bc\cos(A)$
This is done by rearranging the terms of the previous equation.

Answer:

rearranging