Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (-3, -3)

Verified step by step guidance

1. Identify the coordinates of the airplane relative to the radar station at the origin: \((-3, -3)\).
2. Calculate the angle \(\theta\) that the line from the origin to the point makes with the positive x-axis using the arctangent function: \(\theta = \arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{-3}{-3}\right)\).
3. Determine the quadrant of the point \((-3, -3)\) to correctly adjust the angle \(\theta\) because arctangent alone only gives values between \(-90^\circ\) and \(90^\circ\). Since both \(x\) and \(y\) are negative, the point lies in the third quadrant, so add \(180^\circ\) to the angle obtained from arctangent to get the correct direction from the positive x-axis.
4. Express the bearing in the first method (the standard compass bearing) by measuring the angle clockwise from the north (positive y-axis). To do this, convert the angle from the positive x-axis to the angle from the positive y-axis by using the relationship: \(\text{bearing} = 90^\circ + \theta\) (adjusted for quadrant).
5. Express the bearing in the second method (the quadrant bearing) by stating the angle as degrees east or west of north or south. Since the point is in the southwest quadrant, the bearing will be expressed as an angle west of south, calculated by finding the acute angle between the line and the south direction.

Key Concepts

Bearing and Its Methods

Bearing is a way to express direction relative to a reference point, usually north. The two common methods are the compass bearing, which measures angles clockwise from north, and the quadrant bearing, which expresses direction as an angle east or west of north or south. Understanding both methods is essential to convert coordinates into bearings.

Rectangular Coordinate System and Position Vectors

A rectangular coordinate system uses x and y axes to locate points in a plane. The position of an object, like an airplane, can be represented as a vector from the origin (observer) to the point (x, y). This vector helps determine the direction and distance of the object relative to the observer.

Calculating Angles Using Inverse Trigonometric Functions

To find the bearing from coordinates, the angle between the position vector and a reference axis is calculated using inverse tangent (arctan) or other inverse trig functions. Correctly interpreting the quadrant of the point is crucial to determine the accurate bearing angle.