A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?
We're asked to find the total displacement, both the magnitude and direction, of the ship after it leaves the port with the given conditions.
First, I'll explain what a bearing is.
A bearing is NOT a regular angle measure; normally, angles are measured anticlockwise from the positive
#x#-axis, but bearing angles are measured clockwise from the positive #y#-axis.
Therefore, a bearing of
#34.0^"o"#indicates that this is an angle #90.0^"o" - 34.0^"o" = color(red)(56.0^"o"#measured normally. We'll use this angle in our calculations.
We're given that the first displacement is
Our second displacement is a simple
To find the total displacement from the port, we'll add these two vectors' components and use the distance formula:
The direction of the displacement vector is given by
so the angle is then
The question asked for the bearing angle, which is just this angle subtracted from
Bearing is a clockwise angle measured from due North. This is a problem, because all of the trigonometric functions are referenced to a counterclockwise angle measured from East.
A bearing of
The (x,y) values for the position of the ship after completing its first heading are:
The trigonometric angle for the second heading is
The (x,y) values for the position of the ship after completing its second heading is:
The distance from port is:
Its trigonometric angle is:
The bearing angle is:
Let say the distance of ship from port after travelled to the east
and the angle between a bearing of
we use consine formula to find
we use sinus formula to find the angle of displacement to east, let say
therefore it bearing from the port