# Trigonometry : Bearing

## Example Questions

### Example Question #1 : Bearing

Which of the following could represent an aeronautical bearing of ?

Explanation:

The correct image depicting an aeronautical bearing of  is

This image begins at north, and moves  clockwise from it.

Three of the given incorrect answers depict . The fourth incorrect answer does not represent a standard bearing convention as it is neither an acute angle, nor in the clockwise direction. That incorrect answer looks like:

### Example Question #2 : Bearing

Which of the following diagrams could show a bearing of ?

Explanation:

The bearing of a point B from a point A in a horizontal plane is defined as the acute angle made by the ray drawn from A through B with the north-south line through A. The bearing is read from the north or south line toward the east or west. Bearing is typically only represented in degrees (or degrees and minutes) rather than radians. To find , start in the south direction, then move  towards the west:

The other incorrect answer choices provided depict , and .

### Example Question #3 : Bearing

The following diagram could represent which one of these practical scenarios?

A helicopter traveling  at  miles per hour for  hours

A race car traveling  at  miles per hour

An airplane traveling  at  miles per hour

A motorboat traveling  at  miles per hour for  hours

A motorboat traveling  at  miles per hour for  hours

Explanation:

This question and its answer choices give you a few clues to work with. First, we need to identify the bearing angle being shown. The options in the answer choices are either , or . Because the angle begins in the south direction and moves  towards the west, the correct bearing is . That means only two of the answer choices could be correct. We now need to understand how the  miles per hour corresponds to the problem. Notice that there is no answer choice that has the bearing of  and velocity of  miles per hour. Rather, we need to choose between  miles per hour for  hours or  miles per hour for  hours. Because  miles per hour for  hours corresponds to  (whereas the other option corresponds to only ), the correct answer is "A motorboat traveling  at  miles per hour for  hours."

### Example Question #4 : Bearing

A ship moves in the direction  at a speed of  miles per hour for  hours. How far south and how far east is the ship from its starting position?

181.87 miles south and 105 miles east

30.31 miles south and 17.5 miles east

17.5 miles south and 30.31 miles east

105 miles south and 181.87 miles east

181.87 miles south and 105 miles east

Explanation:

First, let's set up a diagram using the given information. This looks like this:

Next, let's convert this info into a triangle so that we can use trigonometry to solve the problem. We need to calculate that the ship going  miles per hour for  hours will have traveled  miles.

Now we can use trigonometry to determine the missing sides, s and e.

Therefore the ship has travelled 181.87 miles south and 105 miles east.

### Example Question #5 : Bearing

An airplane is traveling at a bearing of  from north for 330 kilometers. How far south and how far east is the plane from its starting point?

The airplane is 139.46 km south of its starting point and 299.08 km east of its starting point.

The airplane is 299.08 km south of its starting point and 139.46 km east of its starting point.

The airplane is 139.46 km south of its starting point and 707.69 km east of its starting point.

The airplane is 707.69 km south of its starting point and 139.46 km east of its starting point.

The airplane is 139.46 km south of its starting point and 299.08 km east of its starting point.

Explanation:

First, let's incorporate the given information into a diagram. Start by labelling the plane's bearing of  along with its velocity 330km. Next, draw a line segment to complete the triangle and determine the measures of the angles of the triangle. We can determine the angle , we constructed the diagram such that there is a right angle, and finally we can find the third angle by taking

The question is asking us how far south and how far east the plane is from its starting point, so we need to now use trigonometry to determine the lengths of the missing sides of the triangle. We will call these sides s for the southward distance and e for the eastward distance.

km

km

Therefore the airplane is 139.46 km south of its starting point and 299.08 km east of its starting point.

### Example Question #6 : Bearing

Three ships are positioned in the following way: Ship A is 240 miles due west of Ship C, and Ship B is due south of Ship C. Ship B bears  from Ship A. How far is Ship B from Ship A? How far is Ship B from Ship C? What is the bearing from Ship A to Ship B?

Distance from Ship A to Ship B: 594.02 miles

Distance from Ship B to Ship C: 640.67 miles

Bearing from Ship B to Ship A:

Distance from Ship A to Ship B: 640.67 miles

Distance from Ship B to Ship C: 594.02 miles

Bearing from Ship B to Ship A:

Distance from Ship A to Ship B: 594.02 miles

Distance from Ship B to Ship C: 640.67 miles

Bearing from Ship B to Ship A:

Distance from Ship A to Ship B: 640.67 miles

Distance from Ship B to Ship C: 594.02 miles

Bearing from Ship B to Ship A:

Distance from Ship A to Ship B: 640.67 miles

Distance from Ship B to Ship C: 594.02 miles

Bearing from Ship B to Ship A:

Explanation:

To solve this problem, begin with a diagram, and label all known information. We know that we can label two angles as  and one length as 240. Then, by deductive reasoning, we can label another angle as  because .

Next, we need to use trigonometry to find the answers to each question we're being asked. To find the distance AB, set up

Next, we can find the distance BC. There are two ways to do this since we know two angles of the triangle, but either way you need to use the tangent function.

Finally, we are asked to find the bearing from Ship B to A. Be careful, because in the initial problem we are given the Bearing from Ship A to Ship B. While the angle will remain the same, the direction is different because we are starting at a different initial point (B instead of A). With B as your starting point, A is north and west. Therefore the bearing from Ship B to Ship A is .

Therefore the correct distance between Ships A and B is 640.67 miles, the distance between ships B and C is 594.02 miles, and the bearing from Ship B to Ship A is  .

### Example Question #7 : Bearing

Three cruise ships are situated as follows: Sea Terraformer is 200 miles due north of Wave Catcher, and Island Pioneer is 345 miles due east of Wave Catcher. What is the bearing from Island Pioneer to Sea Terraformer, and what is the bearing from Sea Terraformer to Island Pioneer?

Bearing from Island Pioneer to Sea Terraformer:

Bearing from Sea Terraformer to Island Pioneer:

Bearing from Island Pioneer to Sea Terraformer:

Bearing from Sea Terraformer to Island Pioneer:

Bearing from Island Pioneer to Sea Terraformer:

Bearing from Sea Terraformer to Island Pioneer:

Bearing from Island Pioneer to Sea Terraformer:

Bearing from Sea Terraformer to Island Pioneer:

Bearing from Island Pioneer to Sea Terraformer:

Bearing from Sea Terraformer to Island Pioneer:

Explanation:

Begin by diagramming the given information; you'll see that the three ships create a right triangle. To solve the question, we need to find the unknown angles of the triangle, then frame our answers as the proper bearings.

First let's solve for the topmost angle in the diagram, which we'll call .

By alternate interior angles, the other diagrammed angle will also be equal to .

Now, put your cursor on Island Pioneer, and see if you need to move north or south, and then east or west to get to Sea Terraformer. You need to move north, then west. Therefore the bearing between Island Pioneer and Sea Terraformer is . Now put your cursor on Sea Terraformer and complete the same process to get to Island Pioneer; you'll go south, then east. Therefore the bearing from Sea Terraformer to Island Pioneer is .