An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 41 steps at an angle 45∘ north of east, then 81 steps at an angle 60∘ north of west, then 49 steps due south. Assume his steps all have equal length. Save him from becoming hopelessly lost in the jungle by giving him the displacement, calculated using the method of components, that will return him to his hut.

What is the direction of the displacement that will return the explorer to its hut? degrees EAST OF SOUTH
I got the displacement =51 steps.
I keep getting 75 degrees for the angle but it is south of east and I need the answer to be east of south.

North: 41sin45+81sin60+49(-1)

check that. then do the East the same way. Notice on the N of West, that will be a negative east.

To find the displacement that will return the explorer to his hut, we need to add up the individual displacements caused by each set of steps taken.

First, let's break down the steps taken by the explorer:

41 steps at an angle 45∘ north of east can be broken down into two components:
- North component: 41 * sin(45) = 41 * √(2)/2 = 41√2/2
- East component: 41 * cos(45) = 41 * √(2)/2 = 41√2/2

81 steps at an angle 60∘ north of west can also be broken down into two components:
- North component: 81 * sin(60) = 81 * √(3)/2 = 81√3/2
- West component: 81 * cos(60) = 81 * 1/2 = 81/2

Finally, 49 steps due south has a south component of -49.

Now, let's add up the components to find the overall displacement:

North component: (41√2/2) + (81√3/2) + (-49) = 20.5√2 + 40.5√3 - 49
East component: (41√2/2) - (81/2) = 20.5√2 - 40.5

To find the magnitude of the displacement, we can use the Pythagorean theorem:

Magnitude = √[(North component)^2 + (East component)^2]
= √[(20.5√2 + 40.5√3 - 49)^2 + (20.5√2 - 40.5)^2]

Calculating this value gives the magnitude of the displacement as approximately 51 steps.

To determine the direction of the displacement, we can use the tangent function:

Tangent of the angle = (East component) / (North component)
= (20.5√2 - 40.5) / (20.5√2 + 40.5√3 - 49)

Calculating this value gives us a tangent of approximately 0.961.

Now, to find the angle, we can use the inverse tangent function (arctan) on the tangent value:

Angle = arctan(0.961) = 42.47°

However, this angle is measured from the positive x-axis in the counterclockwise direction. Since we want the angle to be east of south, we need to convert it by subtracting it from 90°:

Angle = 90° - 42.47° ≈ 47.53°

Therefore, the direction of the displacement that will return the explorer to his hut is approximately 47.53 degrees east of south.