Understanding Bearings in Directional Problems

Date: 01/14/2004 at 04:48:49
From: salem
Subject: How to find a distance and bearing using trigonometry

A boat sails 10km from a harbor H on a bearing of S30 degree E.  It 
then sails 15 km on a bearing of N20 degree E.  How far is the boat
from H?  What is the bearing from H?

I drew a triangle and entered in 30 and 20 degrees.  Since a triangle
is made up of 180 degrees, the third angle must be 130 degree since
180 - (30 + 20) equals 130.  Then, I used cosine to find the distance
and then the sine to find the bearing.  But the answer seemed to be wrong.

Date: 01/14/2004 at 09:03:26
From: Doctor Rick
Subject: Re: How to find a distance and bearing using trigonometry

Hi, Salem.

Bearings are not the same as interior angles of a triangle.  The 
bearings are all measured as angles from north or south, not from the 
adjacent side of the triangle.  Here is a picture (not great, but I 
hope it will give the idea):

                                           E
                                          /
                                         /
                                        /
                                       /
          H                           /
          |\                         /
          | \                       /
          |  \                     /
          |   \                   / 15 km
          |    \                 /
          |S30E \               /
          |      \ 10 km       /
          S       \     N     /
                   \    |N20E/
                    \   |   /
                     \  |  /
                      \ | /
                       \|/
                        T

The boat sails 10 km from H to T; the direction is 30 degrees to the 
east (counterclockwise) of south.  At T it turns to a bearing 20 
degrees east of north, and sails 15 km to the end point E.

Now, how can you find angle HTE?  Hint: SH and TN are parallel.

Once you know angle HTE you can use the Law of Cosines to find the
length of HE, then the Law of Sines or Law of Cosines to find angle
EHT.  Remember that to answer the question correctly, you will need to
interpret angle EHT in terms of a bearing from H to E.

Good luck, and write back if you need more help.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/15/2004 at 01:18:53
From: salem
Subject: Thank you (How to find a distance and bearing using trigonometry)

Dear Dr. Rick,

I would like to thank you wholeheartedly for your assistance.

Best regards,

Salem