From city a to city b, a plane flies 650 miles at a bearing of 48 degrees. from city b to city c, the plane flies 810 miles at a bearing of 115 degrees. find the distance from city a to city c and the bearing from city a to city c.

You need to find the angle ABC.
 length AB = 650
 length BC = 810
 angle ABC then you can determine length AC from side-angle-side fornula. 
. Draw your x-axis and y-axis and city A will be at the origin.
 All angles are measured counter-clockwise from the +ve x-axis.
 Now draw a medium length line from the origin at about 45 degrees to approximate the 48 degree bearing.
  put another point about halfway up the line. This will be city B.
draw a line starting at point B at an angle that looks like it's about 135 degrees from the positive x-axis and put point C on this line so that BC is little bigger than AB.
Note where this line meets the line where cities A and B are on there are two angles. One of them is Angle ABC, and the other one when added to Angle ABC = 180 degrees.
We know the value of this other angle since it's just 115 - 48 = 67 degrees. So angle ABC is 180-67 = 113 degrees. now you connect points A and C to complete the triangle formed by the three cities. Now we can determine length AC from the law of cosines: If x, y are the lengths of the known sides and Ang is the angle between them then the third side z is: 

z = sqrt(x^2 +y^2 - 2xycos(Ang)) 

AC = sqrt(650^2 + 810^2 - 2*650*810*cos(113)) = 1220.67 mi 

You can see how to do this if you draw a horizontal line passing through C. If we determine angle ACB and also the angle between BC and the horizontal line passing through C then we will have our bearing if we subtract these two angles from 360 degrees. The one angle is just 180-115 = 65 degrees.
 We can get angle ACB from the law of sines. 

We have: AC/sin(ABC) = AB/sin(ACB) 

AC = 1220.67; AB = 650; ABC = 113 degrees 

So sin(ACB) = (650/1220.67)*sin(113)= 0.490 

Using the arcsine function we have ACB = 29.35 degrees 

Adding the two angles we get 29.35+65= 94.35 degrees 

The bearing is just 360-94.35 = about 265.6 degrees from the +ve x-axis.hope this helps