Law of Cosine Word Problems

**jordangiscool** · July 12th 2009, 05:01 PM

A plane flies 810 miles from A to B with a bearing of N 75 degrees E. Then it flies 648 miles from B to C with a bearing of N 32 degrees E. Find the straight-line distance and bearing from C to A

**sa-ri-ga-ma** · July 13th 2009, 03:20 AM

Originally Posted by jordangiscool

A plane flies 810 miles from A to B with a bearing of N 75 degrees E. Then it flies 648 miles from B to C with a bearing of N 32 degrees E. Find the straight-line distance and bearing from C to A

The angle between AB and AC is 180 - 75 + 32 degrees

**Soroban** · July 13th 2009, 06:08 AM

Hello, jordangiscool!

A plane flies 810 miles from A to B with a bearing of N 75° E.
Then it flies 648 miles from B to C with a bearing of N 32° E.
Find the straight-line distance and bearing from C to A

Code:

                          Q     o C
                          :    *
                          :32°* 
                          :  * 
                          : *
      P              105° |*
      :                   o B
      :               *   :
      :           *   75° :
      : 75°   *           :
      :   *               R
    A o

The plane flies from $A$ to $B\!:\;\;AB \,=\,810$
. . $\angle PAB \,=\, 75^o \,=\,\angle ABR \quad\Rightarrow\quad \angle ABQ \,=\,105^o$

Then it flies from $B$ to $C\!:\;\;BC \,=\,648$
. . $\angle QBC \,=\,32^o \quad\Rightarrow\quad \angle ABC \,=\,137^o$

Draw line segment $AC.$

We have . $\Delta ABC\!:\;\;AB = 810,\;BC = 648,\;\angle B = 137^o$

Law of Cosines: . $AC^2 \;=\;AB^2 + BC^2 - 2(AB)(BC)\cos B$

Got it?

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