Question

what compass heading represents 50° north of east?
?°
hint: compass headings are measured from north, going clockwise.

Explanation:

Step1: Understand Compass Heading

Compass headings are measured from north, clockwise. East is 90° from north (since north to east is a right angle, 90°).

Step2: Calculate 50° North of East

If we are 50° north of east, we start from north and move towards east, but adjust for the angle north of east. Wait, actually, north to east is 90°, so 50° north of east means from north, we move towards east, but the angle from north is 90° - 50°? Wait no, wait. Wait, "north of east" means the angle between the east direction and the line is 50° towards north. So from the east axis, we go 50° towards north. But compass heading is from north, clockwise. So north is 0°, east is 90°, south is 180°, west is 270°. So if we are 50° north of east, that means the angle from north is 90° - 50° = 40°? Wait no, wait. Let's think again. If you are facing east (90° from north), and then turn 50° towards north, your new heading from north (clockwise) would be 90° - 50° = 40°? Wait, no, that can't be. Wait, no: if you are at east (90°), and you go 50° towards north, that means your direction is 50° above the east axis towards north. So the angle from north is 90° - 50° = 40°? Wait, no, let's draw a diagram. North is 0°, east is 90° (right angle). So "north of east" means the angle between the east line and the vector is 50° towards north. So the vector is 50° above the east axis. So the angle from north (clockwise) is 90° - 50° = 40°? Wait, no, that's not right. Wait, no: if you are at north (0°), and you turn towards east, 90° is east. So if you are 50° north of east, that means the angle between the east direction and your direction is 50° towards north. So your direction is east plus 50° towards north. So from the north axis, clockwise, it's 90° - 50° = 40°? Wait, no, I think I messed up. Wait, let's take an example. If it's 0° north of east, that's east (90°). If it's 90° north of east, that's north (0°). So 50° north of east would be 90° - 50° = 40° from north? Wait, no, that would be 40° towards east from north. Wait, no, I'm confused. Wait, the hint says compass headings are measured from north, going clockwise. So north is 0°, east is 90°, south is 180°, west is 270°. So "50° north of east" means the angle between the east direction and the path is 50° towards north. So the path is 50° above the east axis. So to find the compass heading (from north, clockwise), we need to calculate how many degrees from north (clockwise) is this direction. So east is 90° from north. If we are 50° north of east, that means we are 50° towards north from east. So the angle from north is 90° - 50° = 40°? Wait, that seems low. Wait, no, maybe I have it backwards. Let's think of "north of east" as the angle between the north direction and the path is 50° towards east? No, the term is "north of east", which means the reference is east, and the angle is towards north. So "A° north of east" means: start at east, turn A° towards north. So the direction is east + A° towards north. So the angle from north (clockwise) is 90° - A°. So for A = 50°, it's 90° - 50° = 40°? Wait, but that would be 40° from north, towards east. But that's 40° east of north, not north of east. Oh! Wait, here's the mistake. "North of east" and "east of north" are different. "North of east" means the angle is measured from the east axis towards north. "East of north" is measured from the north axis towards east. So "50° north of east" is 50° above the east axis towards north. "50° east of north" is 50° to the east of the north axis. So in this problem, it's 50° north of east. So to find the compass heading (from north, clockwise), we need to find the angle from north, clockwise, to this direction. So east is 90° from north. If we are 50° north of east, that means the direction is 50° above the east axis. So the angle from north is 90° - 50° = 40°? Wait, no, that's east of north. Wait, I think I mixed up the terms. Let's clarify:

- "East of north" (e.g., 50° east of north): angle from north axis towards east is 50°, so compass heading is 50°.
- "North of east" (e.g., 50° north of east): angle from east axis towards north is 50°, so compass heading is 90° - 50° = 40°? No, that can't be. Wait, no, if you are 50° north of east, your direction is such that the angle between the east line and your line is 50° towards north. So the line is 50° above the east line. So the angle from north (clockwise) is 90° - 50° = 40°? But that would be 40° east of north. Wait, this is confusing. Wait, let's use coordinates. Let's set north as the positive y-axis, east as the positive x-axis. So a direction that is 50° north of east would have an angle of 90° - 50° = 40° from the positive y-axis (north) towards the positive x-axis (east). Wait, no, in standard position (from positive x-axis, counterclockwise), but compass is from positive y-axis (north), clockwise. So in standard position (x-axis east, y-axis north), the angle from the x-axis (east) towards y-axis (north) is 50°, so the angle in standard position (counterclockwise from x-axis) is 90° - 50° = 40°? No, standard position is counterclockwise from x-axis (east). So if it's 50° north of east, that's 50° above the x-axis (east) towards y-axis (north), so the angle in standard position is 50°? Wait, no, standard position is counterclockwise from x-axis. So x-axis is 0°, y-axis is 90° (counterclockwise). So 50° north of east is 50° counterclockwise from x-axis (east) towards y-axis (north), so angle is 50° in standard position. But compass heading is clockwise from y-axis (north). So to convert from standard position (counterclockwise from x-axis) to compass heading (clockwise from y-axis), we can use the formula: compass heading = 90° - standard angle. So if standard angle is 50°, compass heading is 90° - 50° = 40°? Wait, that matches. So yes, 50° north of east is 40° compass heading (from north, clockwise). Wait, but let's test with east: east is 90° in standard position (counterclockwise from x-axis), so compass heading is 90° - 90° = 0°? No, that's not right. Wait, no, east is 90° from north (clockwise), so compass heading is 90°. So my formula is wrong. Wait, let's re-express:

Compass heading (H) is measured clockwise from north (y-axis). So:

- North: H = 0°
- East: H = 90° (since it's 90° clockwise from north to east)
- South: H = 180°
- West: H = 270°

Standard position (θ) is measured counterclockwise from east (x-axis). So:

- North: θ = 90° (counterclockwise from x-axis to y-axis)
- East: θ = 0°
- South: θ = 270°
- West: θ = 180°

To convert from standard position (θ) to compass heading (H):

H = 90° - θ (if θ is between 0° and 90°)

Wait, let's test:

- East: θ = 0°, H = 90° - 0° = 90° (correct)
- North: θ = 90°, H = 90° - 90° = 0° (correct)
- Northeast (45° north of east): θ = 45°, H = 90° - 45° = 45° (correct, because 45° north of east is 45° from east towards north, so compass heading is 45°? Wait, no, 45° north of east should be 45° from east towards north, so from north, clockwise, it's 90° - 45° = 45°, which is correct. So yes, that formula works.

So for 50° north of east, θ (standard position, counterclockwise from east) is 50°, so compass heading H = 90° - 50° = 40°? Wait, no, that would be 40° from north, clockwise. But let's check with another example: 90° north of east is north, which should be 0° compass heading. Using the formula, θ = 90°, H = 90° - 90° = 0° (correct). 0° north of east is east, H = 90° - 0° = 90° (correct). So yes, the formula works. So 50° north of east: θ = 50°, H = 90° - 50° = 40°? Wait, no, that can't be. Wait, no, I think I have the standard position wrong. Wait, standard position is counterclockwise from the positive x-axis (east). So if you are 50° north of east, that means the angle between the east (x-axis) and your direction is 50° towards north (y-axis). So in standard position, that's 50° counterclockwise from x-axis (east) towards y-axis (north). So θ = 50°, and compass heading (clockwise from y-axis) is 90° - θ = 40°. So yes, that's correct. So the compass heading is 40°? Wait, no, wait, that seems too low. Wait, no, let's take a concrete example. If you are facing east (90° compass heading), and you turn 50° towards north, your new direction is 50° above the east line towards north. So from the north line (y-axis), clockwise, how many degrees is that? From north (0°), clockwise to east is 90°. So if you are 50° above east towards north, that's 90° - 50° = 40° from north. So yes, the compass heading is 40°. Wait, but I think I made a mistake earlier. Wait, no, let's confirm with the hint: "Compass headings are measured from north, going clockwise." So north is 0°, east is 90°, south is 180°, west is 270°. So "50° north of east" means the angle between the east direction and the path is 50° towards north. So the path is 50° above the east axis. So the angle from north (clockwise) is 90° - 50° = 40°. So the compass heading is 40°.

Step3: Verify

If we have 0° north of east, that's east (90°), which matches 90° - 0° = 90°. If we have 90° north of east, that's north (0°), which matches 90° - 90° = 0°. So 50° north of east should be 90° - 50° = 40°. So the compass heading is 40 degrees.

Answer:

40