Trig

[shooterman]

A fire is sighted from two ranger stations that are 5000m apart. The angles of observation to the fire measure 52 degrees from one station and 1 degrees form the other station. Find the distance along the line of sight to the fire from the closer of the two stations.
So the hyptouse is the line of sight

[attachment=0:3s9qvj1v]th_untitled.jpg[/attachment:3s9qvj1v]

How should i start?
By finding the angles? if so, which other than the 90 degree right angle

 

[soroban]

Hello, shooterman!

A fire is sighted from two ranger stations that are 5000 m apart.
The angles of observation to the fire measure 52 degrees from one station and 41 degrees from the other station.
Find the distance along the line of sight to the fire from the closer of the two stations.

    C *
       *  *
          *   *
            *  11 *
           x  *       *
                *         *
               52 * 128    41 *
          *- - - - - * - - - - - - *
          D          A    5000     B

\(\displaystyle \text{The ranger stations are }A\text{ and }B\!:\;AB = 5000\)

\(\displaystyle \text{The fire is at }C\!:\;\anbgle CBA = 41^o\)

\(\displaystyle \angle CAD = 52^o \quad\Rightarrow\quad \angle CAB = 128^o\)

\(\displaystyle \text{Hence: }\:\angle ACB = 180^o - 128^o - 41^o \:=\:11^o\)

\(\displaystyle \text{Let }x \,=\,AC\)


\(\displaystyle \text{Law of Sines: }\;\frac{x}{\sin41^o} \:=\:\frac{5000}{\sin11^o}\)

\(\displaystyle \text{Therefore: }\;x \;=\;\frac{5000\sin41^o}{\sin11^o} \;\approx\;17,\!191.5\text{ m}\)

 

[shooterman]

Thanks for the great explaination and oh nice picture

 

[mmm4444bot]

Thanks for the great explaination …

Yes; I think that it's exceptional.