Answer:
0.69473828 radians east of north.
Step-by-step explanation:
Interpreting the information given on a Cartesian plane, the position of the ship will be (1,0) and that of the rock will be (7,5). Connecting the point (1,0) and (7,5) with a line and dropping a perpendicular line from point (7,5) to the x-axis to intersect the x-axis at (7,0), will form a right-triangle with base length 6 units and a height of 5 units.
Finding the base length from points (1,0) and (7,0)
d=√(x₂-x₁)²+(y₂-y₁)²
d=√(7-1)²
d= √6²
d=6 units
Finding the height from points (7,5) and (7,0)
d=√(x₂-x₁)²+(y₂-y₁)²
d=√(7-7)² + (0-5)²
d=√-5²
d=√25
d=5 units
So you now have a right triangle with base length 6 units and height 5 units.
To get the bearing of rock from ship position you apply the formula for tangent of an angle
Tan Ф = length of opposite side/length of adjustment side
TanФ=5/6
Tan Ф =0.83333333333
Tan⁻¹ (0.83333333333) =0.69473828 radians east of north.
By using the inverse tangent function we will see that the bearing is 0.876 radians east of north,
For a set of two points (x₁, y₁) and (x₂, y₂), the bearing of the first point with respect to the second one is given by:
θ = Atan( (y₂ - y₁)/(x₂ - x₁))
Where Atan(x) is the inverse tangent function.
In this case, the points are:
(1, 0) and (7, 5), so the angle will be:
θ = Atan( (5 - 0)/(7 - 1)) = Atan(5/6) = 39.8°
The bearing (as we got it) is measured from the positive x-axis, in this case, the east, so the angle measured from the north. the positive y-axis will be:
90° - 39.8° = 50.2°
Now we need to convert this to radians. Remember that:
180° = 3.14 rad
1 = (3.14 rad)/(180°)
Multiplying our measure by this we get:
50.2° = 50.2°*(3.14 rad)/(180°) = 0.876 rad
So the bearing is 0.876 radians east of north,
If you want to learn more about trigonometry, you can read:
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