1. We have three different ways we can give direction when we are navigating around the world. These directions are all based off a compass. The three ways we can give directions/bearings are as follows: The big angle based of the compass The angle based of the “X” axis (EAST & WEST) The angle based of the “Y” axis (NORTH & SOUTH)

2. Example: 140 0 Use the following steps to find the other two ways to give directions. 1. Draw a compass. 2. Draw a line on your compass that represents that angle or compass bearing. You must start from NORTH. 3. You will figure out what angle the line makes with the “X” axis. To find the small angle we will need to subtract bigger angle from the smaller angle. In the end this angle MUST be less then 90 0 140 0 – 90 0 = 50 0 That angle is formed with the “X” axis, so we base the direction as from the “X” axis. When we write the direction we will write them backwards. This means that the axis you are measuring your angle FROM goes LAST. 50 0 S of E First we need to figure out what “X” axis angle we need to deal with. In this case we are going to use the EAST axis (90 0 )

3. To find our final angle, subtract the “X” axis angle from 90 0 90 0 – 50 0 = 40 0 40 0 E of S Example: 140 0 (cont’d) 4. This line can also be measured from the “Y” Axis. The easiest way to do this, once we know our angle to the “X” Axis, is to subtract. We know that the angle between South and West is 90 o. That angle is formed with the “Y” axis, so we base the direction as from the “Y” axis. When we write the direction we will write them backwards. This means that the axis you are measuring FROM goes LAST. In this case that is the South axis.

4. 50 0 S of E 140 0 40 0 E of S So, the line drawn on the compass can be represented 3 ways. All three of these represent the same direction or compass bearing. Example: 140 0

5. Example: 335 0 Use the following steps to find the other two ways to give directions. 1. Draw a compass. 2. Draw a line on your compass that represents that angle or compass bearing. You must start from NORTH. 3. You will figure out what angle the line makes with the “X” axis. To find the small angle we will need to subtract bigger angle from the smaller angle. In the end this angle MUST be less them 90 0 335 0 – 270 0 = 65 0 That angle is formed with the “X” axis, so we base the direction as from the “X” axis. When we write the direction we will write them backwards. This means that the axis you are measuring your angle FROM goes LAST. 65 0 N of W First we need to figure out what “X” axis angle we need to deal with. In this case we are going to use the WEST axis (270 0 )

6. To find our final angle, subtract the “X” axis angle from 90 0 90 0 – 65 0 = 25 0 25 0 W of N Example: 335 0 (cont’d) 4. This line can also be measured from the “Y” Axis. The easiest way to do this, once we know our angle to the “X” Axis, is to subtract. We know that the angle between NORTH and WEST is 90 o. That angle is formed with the “Y” axis, so we base the direction as from the “Y” axis. When we write the direction we will write them backwards. This means that the axis you are measuring FROM goes LAST.

7. 65 0 N of W 335 0 25 0 W of N So, the line drawn on the compass can be represented 3 ways. All three of these represent the same direction or compass bearing. Example: 335 0

8. Example: 20 0 S of W Use the following steps to find the other two ways to give directions. 1. Draw a compass. 3. Draw a line on your compass that represents that angle or compass bearing. To draw in the line, the angle we need to measure starts from the WEST axis (“X”) and move towards the SOUTH axis (“Y”) 20 0. To find the other small angle, subtract the first small angle from 90 0 90 0 – 20 0 = 70 0 To get the direction part all you need to do is FLIP the order for the directions from the original problem. 70 0 W of S 2. Remember that we read compass bearings backwards. So, the bearing above literally means: “Start on the WEST axis and then move toward the SOUTH axis 20 o.

9. 4. We can find the large angle two different ways. If we have a compass all we need to do is read the angle off the diagram. If we don’t have a diagram we will need to do some math. You will have two choices when you are using this method. We know that the large angle will need to be between 180 0 (WEST) & 270 0 (SOUTH) due to the fact that we are in the SOUTH – WEST quadrant. We will use the two angles that we found previously to find the large angle. This can be done by either adding to 180 0 (SOUTH) or subtracting from 270 0 (WEST) We know that we are 70 0 past the 180 0 (SOUTH), so we can add: 180 0 + 70 0 to get 250 0. Remember that we need to start from the NORTH axis. We also know that we are 20 0 short of the 270 0 (WEST), so we can subtract: 270 0 - 20 0 to get 250 0. In both cases we get the same answer. 250 o Example: 20 0 S of W (cont’d)

10. So, the line drawn on the compass can be represented 3 ways. All three of these represent the same compass bearing. 20 0 S of W 250 0 70 0 W of S Example: 20 0 S of W

11. Example: 35 0 E of N Use the following steps to find the other two ways to give directions. 1. Draw a compass. 3. Draw a line on your compass that represents that angle or compass bearing. To draw in the line, the angle we need to measure starts from the NORTH axis (“Y”) and move towards the EAST axis (“X”) 35 0. To find the other small angle, subtract the first small angle from 90 0 90 0 – 35 0 = 55 0 To get the direction part all you need to do is FLIP the order for the directions from the original problem. 55 0 N of E 2. Remember that we read compass bearings backwards. So, the bearing above literally means: “Start on the NORTH axis and then move toward the EAST axis 35 o.

12. 4. We can find the large angle two different ways. If we have a compass all we need to do is read the angle off the diagram. If we don’t have a diagram we will need to do some math. You will have two choices when you are using this method. We know that the large angle will need to be between 0 0 (NORTH) & 90 0 (EAST) due to the fact that we are in the NORTH – EAST quadrant. We will use the two angles that we found previously to find the large angle. This can be done by either adding to 0 0 (NORTH) or subtracting from 90 0 (EAST) We know that we are 35 0 past the 0 0 (NORTH), so we can add: 0 0 + 35 0 to get 35 0. Remember that we need to start from the NORTH axis. We also know that we are 55 0 short of the 90 0 (EAST), so we can subtract: 90 0 - 55 0 to get 35 0. In both cases we get the same answer. 35 o Example: 20 0 S of W (cont’d)

13. So, the line drawn on the compass can be represented 3 ways. All three of these represent the same compass bearing. 35 0 E of N 35 0 55 0 N of E Example: 20 0 S of W For vectors in the NORTH – EAST quadrant will have two angles that are the same. This is the only quadrant that this will take place in.