The ship is 40 miles south and 20 miles east of port how do you know or calculate what bearing the ship would have to sail back to port

figure the final displacement of the ship from the port.

Add 180° to get the heading back to port.

To determine the bearing the ship would have to sail to return to port, we can use trigonometry to calculate the angle between the ship's current position and the port.

1. First, draw a diagram to visualize the situation. Place the port at the origin (0,0) on a coordinate plane, and plot the ship's current position, which is 40 miles south and 20 miles east of the port.

2. To find the bearing, we need to calculate the angle between the line connecting the ship's position with the port, and the positive x-axis.

3. Apply trigonometry: The tangent of an angle is the ratio of the side opposite the angle to the adjacent side. In this case, the "opposite" side is the distance south (40 miles) and the "adjacent" side is the distance east (20 miles).

4. Use the inverse tangent function (arctan) to find the angle. Calculate the arctan of the ratio:

arctan(40 miles / 20 miles)

This will give you the angle in radians. To convert it to degrees, multiply by 180/π (approximately 57.3).

5. The resulting angle will give you the bearing of the ship in relation to the port. Keep in mind that bearings are typically measured clockwise from the north, so you may need to adjust the result accordingly.

For example, if the arctan calculation gives you an angle of 56 degrees, the bearing would be 360 - 56 = 304 degrees (assuming we are using the standard convention).

To determine the bearing the ship would have to sail back to port, we can use trigonometry. The bearing can be calculated using the tangent function.

1. First, draw a diagram of the situation. Place the port at the origin (0,0) on a coordinate plane, and mark the ship's location 40 miles south and 20 miles east of the port.

2. Next, calculate the angle between the line connecting the ship to the port and the x-axis.

- Let's call the point where the ship is located "S" and the port "P".
- The ship is 40 miles south and 20 miles east, so the coordinates of point S are (20, -40).
- Since the x-axis is horizontal, the angle we need is the angle between the line SP and the positive x-axis.

3. Now, calculate the angle using the inverse tangent function (arctan).

- The tangent of an angle is equal to the ratio of the opposite side (in this case, the y-coordinate of S, which is -40) to the adjacent side (the x-coordinate of S, which is 20).
- So, tangent of the angle = -40/20 = -2.
- Taking the arctan of -2 will give us the angle in degrees.

4. Use a calculator to find the arctan of -2, which is approximately -63.43 degrees.

5. Since we want the bearing back to port, we need to add 180 degrees to the above result to get the bearing in true north.

- -63.43 degrees + 180 degrees = 116.57 degrees.

Therefore, the ship would need to sail on a bearing of approximately 116.57 degrees to return back to port.