1. ## Law of Sines

The bearing from the Pine Knob fire tower to the Colt Station fire tower is N 65 degrees East, and the two towers are 30 km apart. A fire spotted by rangers in each tower has a bearing of N 80 degrees East from Pine Knob and S 70 degrees E from Colt Station. Find the distance of the fire from each tower.

2. ## Re: Law of Sines

Hey alexthesauceboss.

Are the two triangles right angled ones?

3. ## Re: Law of Sines

Hello, alexthesauceboss!

$\displaystyle \text{The bearing from the Pine Knob fire tower to the Colt Station fire tower is }N\,65^o\,E.$
$\displaystyle \text{The two towers are 30 km apart. }\,\text{A fire spotted by rangers in each tower.}$
$\displaystyle \text{It has a bearing of }N\,80^o\,E\text{ from Pine Knob and }S\,70^o\,E\text{ from Colt Station.}$
$\displaystyle \text{Find the distance of the fire from each tower.}$

The diagram looks like this:
Code:
                          C
o
* : *
N               *65o:70o*
:             *     :     *
:        30 *       :       *
:         *         :     30o o F
:       *           :   *
: 65o *           * :
:   * 15o   *       S
: *   *
P o
Pine Knob is at $\displaystyle P$, Colt Station is at $\displaystyle C$.
$\displaystyle \angle N\!PC = 65^o,\;PC = 30.$

The fire is at $\displaystyle F.$
$\displaystyle \angle N\!PF = 80^o \quad\Rightarrow\quad \angle CPF = 15^o.$
$\displaystyle \angle FCS = 70^o \quad\Rightarrow\quad \angle PCF = 135^o \quad\Rightarrow\quad \angle F = 30^o$

Law of Sines:

. . $\displaystyle \frac{CF}{\sin15^o} \:=\:\frac{30}{\sin30^o}$

. . $\displaystyle \frac{PF}{\sin135^o} \:=\:\frac{30}{\sin30^o}$

Go for it!