Math Help - Bearing Problems

1. Bearing Problems

Hello, I have a question on a bearing problem. Here's the problem:

A ship sails a distance of 76 miles in the direction S 42° E and then sails a distance of 90 miles in the direction N 48° E. Find the distance of the ship's final position from its original position and the bearing and azimuth of the ship's final position.

My issue here is that I don't really know how to start the problem. I know how to draw the bearing of S 42° E, but I'm not quite sure what to do with the N 48° E.

I also have another problem that I need help on. I attempted it, but the book has a different answer than the one that I have. Here's the problem:

A DC-9 aircraft leaves Midway Airport from runway 4 RIGHT, whose bearing is N 40° E. After flying for 1/2 mile, the pilot requests permission to turn 90° and head toward the southeast. The permission is granted. After the airplane goes 1 mile in this direction, what bearing should the control tower use to locate the aircraft?

Here is my attempt at solving the problem:

tan(x) = 1/(1/2)
tan(x) = 2
x = tan^-1(2)
x = 63.4°
63.4° - 40° = 23.4°
So my answer was S 23.4° E. However, the book says that the answer is S 76.6° E. Can anyone explain what I did wrong?

2. Re: Bearing Problems

So you've drawn your line segment of 76 miles going on a bearing of S42E. At the end of that segment, draw in your "north", and then measure out 48 degrees east of the north. Then draw another line segment going in that direction for a distance of 90 miles.

3. Re: Bearing Problems

Hello, Smugleaf!

I don't agree with the book's answer.
Also, why did you subtract 40 degrees?

A DC-9 aircraft leaves Midway Airport from a runway whose bearing is N 40° E.
After flying for 1/2 mile, the pilot turns 90° and head toward the southeast.
After going 1 mile in this direction, what bearing should the control tower use to locate the aircraft?

The plane leaves the airport $A$ and flies 0.5 miles to $B.$
. . Angle NAB = $40^o$
It turns $90^o$ and flies 1 mile to $C.$

We want angle NAC.

Code:
    N
:     B
:     o
:    *90*
:40 *     *
:  * 0.5    * 1
: *           *
:*              *
A o                 *
*           *
*     *
o C
tan(/BAC) = 1/(0.5)

/BAC ≈ 63.4o

/NAC = 63.4o + 40o = 103.4o

Bearing: .N 103.4o E