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QuestionFind the bearing of an airplane at (−5,−5)(-5,-5) from the origin. Provide the bearing as a single angle measure.

Studdy Solution

STEP 1

Assumptions1. The observer is located at the origin of a coordinate system (0,0). . The airplane is located at the point (-5,-5).
3. The bearing of the airplane is the angle from the positive y-axis (north) to the line connecting the origin and the airplane.
4. The bearing can be expressed in two ways a single angle measure (0 to360 degrees, clockwise from north), or a direction (,,, W) and an angle measure (0 to90 degrees).

STEP 2

First, we need to find the angle from the positive x-axis (east) to the line connecting the origin and the airplane. This can be done using the arctangent function.
θ=arctan⁡(yx)\theta = \arctan\left(\frac{y}{x}\right)

STEP 3

Now, plug in the given values for x and y to calculate the angle.
θ=arctan⁡(−5−5)\theta = \arctan\left(\frac{-5}{-5}\right)

STEP 4

implify the fraction inside the arctangent function.
θ=arctan⁡(1)\theta = \arctan(1)

STEP 5

Calculate the angle. The arctangent of1 is45 degrees.
θ=45∘\theta =45^\circ

STEP 6

Since the airplane is in the third quadrant (both x and y are negative), we need to add180 degrees to the angle to get the bearing from the positive x-axis.
Bearing from east=θ+180∘\text{Bearing from east} = \theta +180^\circ

STEP 7

Plug in the value for theta to calculate the bearing from east.
Bearing from east=45∘+180∘\text{Bearing from east} =45^\circ +180^\circ

STEP 8

Calculate the bearing from east.
Bearing from east=225∘\text{Bearing from east} =225^\circ

STEP 9

To get the bearing from north, we subtract the bearing from east from360 degrees.
Bearing from north=360∘−Bearing from east\text{Bearing from north} =360^\circ - \text{Bearing from east}

STEP 10

Plug in the value for the bearing from east to calculate the bearing from north.
Bearing from north=360∘−225∘\text{Bearing from north} =360^\circ -225^\circ

STEP 11

Calculate the bearing from north.
Bearing from north=135∘\text{Bearing from north} =135^\circThe bearing of the airplane using a single angle measure is135 degrees from north.

STEP 12

The bearing can also be expressed as a direction (,,, W) and an angle. Since the bearing from north is more than90 degrees, we express the direction as east and subtract90 degrees from the bearing.
Bearing= (Bearing from north−90∘)\text{Bearing} =\, ( \text{Bearing from north} -90^\circ)

STEP 13

Plug in the value for the bearing from north to calculate the bearing.
Bearing= (135∘−90∘)\text{Bearing} =\, (135^\circ -90^\circ)

STEP 14

Calculate the bearing.
Bearing= 45∘\text{Bearing} =\,45^\circThe bearing of the airplane using a direction and an angle is45 degrees.

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