MHB Find the Bearing from A to C & Angle B: Solve Here

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To find the distance and bearing from point A to point C, the law of cosines is applied, with the formula AC = √(470² + 250² - 2(470)(250)cos(165). The angle B, which is crucial for this calculation, is derived from the bearings of the two legs of the journey. The first leg from A to B has a bearing of 25 degrees, and the second leg from B to C has a bearing of 40 degrees, leading to an angle BAC of 165 degrees. The discussion emphasizes the need to clarify the calculation of angle ABC and how it relates to the overall geometry of the problem. Understanding these angles is essential for accurately determining the distance and bearing from A to C.
cbarker1
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Dear everyone,

An airplane flies 470 miles from point $A$ to point $B$ with a bearing of 25 degrees. It then flies from 250 miles from point $B$ to point $C$ with a bearing of 40 degrees. Find the distance and the bearing from A to point C.
Work

Bearing Problem.png

I understand that I need to use law of cosines for the side $b$ which is opposite of the angle $B$. But I have a hard time with find what is the angle $B$ is. I forgot many things from geometry. How to determine the angle from point $A$ to point $C$?

Thanks
Cbarker1
 
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$AC = \sqrt{470^2+250^2 - 2(470)(250)\cos(165)}$

bearing = $25^\circ + m\angle{BAC}$

$\angle{BAC}$ may be found using either the sine or cosine law
 
How did you determine the angle ABC to be 165?
 
bearings.jpg
 
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