Question

what compass heading represents 20° east of south? ?° hint: compass headings are measured from north, going clockwise.

Explanation:

Step1: Understand compass heading

Compass headings are measured from north, clockwise. South is 180° from north (since north is 0°, east is 90°, south is 180°, west is 270°).

Step2: Calculate 20° east of south

To find 20° east of south, we start at south (180°) and move 20° towards east (clockwise). So we add 20° to 180°? Wait, no. Wait, south is 180°, east of south means towards east from south. So from north, clockwise: north (0°), east (90°), south (180°), west (270°). East of south: so from south, turning towards east (which is clockwise direction). So the angle from north would be 180° + 20°? Wait, no. Wait, let's think again. North is 0°, clockwise. So south is 180°, east is 90°, west is 270°. So east of south: the direction is south-east, 20° towards east from south. So the angle from north: starting at north (0°), going clockwise to south is 180°, then going 20° towards east (which is clockwise) from south. So total angle is 180° + 20°? Wait, no, that would be 200°, but wait, east is 90°, so south is 180°, east of south is between south (180°) and east (90°)? No, wait, no: east is 90°, south is 180°, so east of south is towards the east from south, so the angle from north would be 180° - 20°? Wait, no, that can't be. Wait, let's use a diagram. North at top, east right, south bottom, west left. So south is bottom (180°), east is right (90°). So 20° east of south: from the south direction (bottom), turn 20° towards east (right). So the angle from north (top) clockwise: from north (0°) to south (180°) is 180°, then from south, turning 20° towards east (right) is clockwise, so we add 20° to 180°? Wait, no, because east is 90°, which is less than 180°, so that can't be. Wait, I think I made a mistake. Wait, the angle between south and east: south is 180°, east is 90°, so the angle between them is 90° (180° - 90° = 90°). So 20° east of south: the direction is 20° towards east from south, so the angle from north is 180° - 20°? Wait, no, that would be 160°, but that's towards east from south? No, 160° would be between north and south? Wait, no, 0° is north, 90° east, 180° south, 270° west. So 160° is south-east? Wait, no, 180° is south, 90° is east, so 160° is 20° west of south? No, that's confusing. Wait, let's use the hint: compass headings are measured from north, going clockwise. So north is 0°, each degree clockwise increases the heading. So south is 180°, east is 90°, west is 270°. So east of south: the direction is towards the east from south, so in clockwise direction, from north, we go to south (180°), then go 20° towards east (which is clockwise) from south. So the total heading is 180° + 20°? Wait, but east is 90°, so 180° + 20° = 200°, but that's more than 180° and less than 270°? No, 200° is between south (180°) and west (270°)? No, west is 270°, so 200° is south-west? No, that's not right. Wait, I think I messed up the direction. East of south means the angle between the south direction and the east direction is 20°, towards east. So the heading (angle from north, clockwise) would be 180° - 20°? Wait, 180° is south, subtracting 20° towards east (since east is left of south? No, east is right of north, so east is right, south is down. So east of south is down and right, so from north (top), clockwise, south is down (180°), east is right (90°), so east of south is down-right, 20° towards right from down. So the angle from north would be 180° + 20°? Wait, no, that would be 200°, but that's down-left? No, I'm getting confused. Wait, let's calculate the correct heading. The formula for a heading that is θ degrees east of south: the heading is 180° + θ? No, wait, no. Wait, north is 0°, east is 90°, south is 180°, west is 270°. So east of south: the direction is S20°E (south 20° east), which is a standard bearing. The heading (angle from north, clockwise) for S20°E is calculated as 180° - 20° = 160°? Wait, no, that can't be. Wait, let's use the standard method. For a bearing of SθE (south θ degrees east), the heading (angle from north, clockwise) is 180° - θ. Wait, no, let's take an example: S0°E is south, which is 180°, so 180° - 0° = 180°, correct. S90°E is east, which is 90°, so 180° - 90° = 90°, correct. So yes, for SθE, the heading is 180° - θ. Wait, that makes sense. Because S0°E is south (180°), S90°E is east (90°), so the formula is 180° - θ. So for 20° east of south, θ = 20°, so heading is 180° - 20° = 160°? Wait, but that contradicts my earlier thought. Wait, let's draw a circle: 0° north, 90° east, 180° south, 270° west. So S20°E: start at south (180°), turn 20° towards east (which is counter-clockwise? No, compass heading is clockwise from north. So east is clockwise from north, so from south, turning towards east is clockwise? Wait, south is 180°, east is 90°, so from south, turning towards east is counter-clockwise, because 90° is less than 180°. Oh! That's the mistake. Compass heading is clockwise from north, so east is 90° (clockwise from north), south is 180° (clockwise from north), west is 270° (clockwise from north). So from south (180°), turning towards east (which is 90°) is counter-clockwise, because 90° is less than 180°. But compass heading is measured clockwise from north, so we can't turn counter-clockwise. So the correct way is: the angle between the north direction and the desired direction, measured clockwise. So for 20° east of south, the direction is such that from north, we go clockwise past east (90°), past south (180°), and then 20° towards east? No, that doesn't make sense. Wait, no, east of south means the direction is between south and east, closer to south. So the angle from north, clockwise, would be 180° - 20° = 160°? Wait, 160° is between east (90°) and south (180°)? No, 160° is 160°, which is 70° past east (90° + 70° = 160°), so that's 70° south of east? No, that's not 20° east of south. I think I need to use the correct formula for compass headings. The formula for a heading that is α degrees east of south is: heading = 180° + α? No, that would be 200°, which is 20° west of south? No, west of south would be 180° + 20° = 200°, because west is clockwise from south. Wait, yes! West is clockwise from south, east is counter-clockwise from south. But compass heading is clockwise from north, so east of south is actually measured as 180° - 20° = 160°, and west of south is 180° + 20° = 200°. Let's verify with a standard compass: north (0°), east (90°), south (180°), west (270°). So the four quadrants: northeast (0°-90°), southeast (90°-180°), southwest (180°-270°), northwest (270°-360°). So 20° east of south is in the southeast quadrant, between east (90°) and south (180°). So the angle from north, clockwise, would be 180° - 20° = 160°? Wait, 180° is south, so 180° - 20° = 160°, which is 20° east of south? Let's check: from south (180°), moving 20° towards east (counter-clockwise) gives 160°, which is 20° east of south. But since compass heading is clockwise from north, we can also calculate it as 360° - (90° - 20°) = 360° - 70° = 290°? No, that's not right. Wait, I'm really confused. Let's use the hint: compass headings are measured from north, going clockwise. So north is 0°, each degree clockwise increases the heading. So to find 20° east of south, we can think of the direction as follows: south is 180°, east is 90°, so east of south is between south (180°) and east (90°), but since we measure clockwise from north, we have to go from north (0°) clockwise to south (180°), then clockwise to east (90°) is not possible, because 90° is less than 180°. Wait, no, clockwise from north: 0° (north), 90° (east), 180° (south), 270° (west), 360° (north). So the direction 20° east of south is in the southeast quadrant, which is between east (90°) and south (180°). So the angle from north, clockwise, is 180° - 20° = 160°? Wait, 160° is 160° clockwise from north, which is 20° east of south? Let's see: from north (0°), clockwise 160°, which is 160° - 90° = 70° past east (90°), so 70° south of east? No, that's not 20° east of south. I think the correct way is: the bearing S20°E (south 20° east) corresponds to a heading of 180° - 20° = 160°? Wait, no, let's check with a calculator. The formula for converting bearing to heading: for a bearing of SθE, the heading is 180° - θ. For S20°E, heading is 160°. For S0°E (south), heading is 180°, correct. For S90°E (east), heading is 90°, correct. So yes, that works. So 20° east of south is S20°E, which is a heading of 160°? Wait, but that seems contradictory to the clockwise measurement. Wait, no, 160° clockwise from north is 160°, which is 20° east of south? Let's draw it: north at top (0°), east at right (90°), south at bottom (180°), west at left (270°). 160° clockwise from north is 160° towards the bottom-right, which is 20° east of south (since from south (180°), moving 20° towards east (right) would be 160°). Yes, that makes sense. So the correct heading is 180° - 20° = 160°? Wait, no, 180° - 20° is 160°, but if we go clockwise from north, 160° is 160°, which is 20° east of south. Wait, I think I was overcomplicating. The key is: compass heading is clockwise from north. South is 180°, east is 90°, so east of south is towards the east from south, which is a clockwise angle from north of 180° - 20° = 160°? No, wait, no: if you are at south (180°), and turn 20° towards east (which is counter-clockwise, because east is at 90°, which is less than 180°), but compass heading is clockwise, so we have to express the angle as clockwise from north. So the angle between north and the desired direction, clockwise, is 180° + (90° - 20°)? No, that's not right. Wait, let's use the hint: compass headings are measured from north, going clockwise. So to find 20° east of south, we can think of the direction as follows: the south direction is 180° clockwise from north. East of south means that from the south direction, we move 20° towards the east direction (which is clockwise, because east is 90° clockwise from north, but south is 180° clockwise from north, so east is counter-clockwise from south). Wait, this is the confusion: clockwise from north, east is 90°, south is 180°, so east is counter-clockwise from south. Therefore, to measure 20° east of south (towards east from south) in clockwise from north, we have to go from north (0°) clockwise past east (90°), past south (180°), and then 20° towards east? No, that's not possible. I think the correct answer is 180° + 20° = 200°? Wait, no, that would be 20° west of south, because west is clockwise from south. Wait, yes! West is clockwise from south, east is counter-clockwise from south. So 20° east of south is 20° counter-clockwise from south, which is 180° - 20° = 160° clockwise from north. And 20° west of south is 180° + 20° = 200° clockwise from north. So the correct answer is 160°? Wait, but let's check with a simple example. If it's 0° east of south, that's south, which is 180°, so 180° - 0° = 180°, correct. If it's 90° east of south, that's east, which is 90°, so 180° - 90° = 90°, correct. So yes, the formula is heading = 180° - θ for θ degrees east of south. Therefore, for 20° east of south, heading is 180° - 20° = 160°.

Answer:

160