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June 23rd, 2017, 10:32 PM   #1
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How do you solve this bearing problem?

From the top of a 50m observation tower, a fire ranger observes smoke in two locations. One is on a bearing of 40 degrees with an angle of depression of 8 degrees and the other is on a bearing of 205 degrees with an angle of depression of 13 degrees. How far apart are the sources of smoke?

I am very confused about how to incorporate bearings into trigonometry. Please help me. Thanks a lot.
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June 23rd, 2017, 11:22 PM   #2
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can you assume that the fires both occur at altitude 0m?
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June 24th, 2017, 05:00 AM   #3
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Assuming both fires have the same vertical elevation as the base of the tower ...

The horizontal distance from the base of the tower to the first fire is $d_1=\dfrac{50}{\tan(8^\circ)}$

The horizontal distance from the base of the tower to the second fire is $d_2=\dfrac{50}{\tan(13^\circ)}$

the angle between their respective bearings is $205^\circ - 40^\circ =165^\circ$

distance between the two fires, $D = \sqrt{d_1^2+d_2^2 -2d_1d_2\cos(165^\circ)}$
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July 2nd, 2017, 12:08 AM   #4
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Thanks a lot, skeeter.
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