# Law of Sines

Printable View

• Jan 21st 2013, 12:54 PM
alexthesauceboss
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.
• Jan 21st 2013, 06:29 PM
chiro
Re: Law of Sines
Hey alexthesauceboss.

Are the two triangles right angled ones?
• Jan 21st 2013, 08:35 PM
Soroban
Re: Law of Sines
Hello, alexthesauceboss!

Quote:

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

The diagram looks like this:
Code:

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

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

Law of Sines:

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

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

Go for it!