There is a trigonometric identity that reads:

sin(a) / A = sin(b) / B = sin(c) / C

where:

A, B and C are lengths of sides of the triangle, and

a = measure of the angle OPPOSITE side A

b = measure of the angle OPPOSITE side B

c = measure of the angle OPPOSITE side C

a and b will equal 52 and 41, depending on how you labelled them, but c will equal 87, since the three angles of a triangle will ALWAYS total 180.

AND, since the smallest side of the triangle ALWAYS lies OPPOSITE the smallest angle, we can deduce that the side opposite the 41 degree angle is the shortest side.

Therefore, given the equation above:

sin(87) / 5000 = sin(41) / X

where X is the distance between the closer ranger station and the fire.

You can look up the sine of 87 degrees and 41 degrees in a trig table, or use a calculator with trig functions in it. the rest is simple arithmetic.

I hope that helped.

sina/a=sinb/b=sinc/c

In this case the station adjacent to the 52° angle would be the nearest.

180 – (41 + 52) = 87

87° is the angle opposite 5000 m

sin 87 / 5000 = sin 41 / x

0.9986/5000 = 0.6560/x

x = 3284.8 m

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