# 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?

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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:

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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

NOTa regular angle measure; normally, angles are measured anticlockwise from the positive#x# -axis, but bearing angles are measuredclockwisefrom the positive.#y# -axisTherefore, 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

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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

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