Problem 69 - Precalculus : Building Concepts and Connections

Question

Explain why \((r, \theta)\) and \((r, \theta+2 \pi)\) represent the same point in the polar coordinate system.

Step-by-Step Solution

Answer

\((r, \theta)\) and \((r, \theta+2 \pi)\) represent the same point in the polar coordinate system because adding \(2\pi\) to the angle corresponds to one full rotation around the circle, leaving the point's position unchanged.

Step 1: Understand Polar Coordinates

In polar coordinates, any point P in a plane can be represented as \((r, \theta)\), where r is the distance from P to the origin and \(\theta\) is the angle formed by the positive x-axis and the line segment from the origin to P.

Step 2: Add \(2\pi\) to the angle

We consider \((r, \theta+2 \pi)\). This means, we're taking the point as defined by \((r, \theta)\), but adding \(2\pi\) to the angle. But adding \(2\pi\) to the angle corresponds to one full rotation around the circle.

Step 3: Compare the two points

We see that, despite this additional rotation or angle \((r, \theta+2 \pi)\), the point's position relative to the origin does not change. So, \((r, \theta)\) and \((r, \theta+2 \pi)\) both represent the exact same point in the polar coordinate system.

Key Concepts

Polar Coordinate System

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance and an angle. In contrast to the rectangular (or Cartesian) coordinate system which uses horizontal and vertical distances (x and y coordinates) to describe the location of a point, the polar coordinate system uses the distance from a reference point, known as the pole (typically represented as the origin in Cartesian coordinates), and an angle from a reference direction, usually the positive x-axis.

Each point is represented by a pair \( (r, \theta) \), where \( r \) is the radius or the straight-line distance from the origin to the point, and \( \theta \) is the angular coordinate, also known as the polar angle. The polar angle is measured in radians and describes the angle between the reference direction and a line from the origin to the point. Unlike Cartesian coordinates, where (x, y) represents a unique point, polar coordinates can sometimes return to the same point with different values of \( \theta \) by multiple full rotations, expressed as \( \theta + 2\pi n \) where n is an integer.

Radians

Radians are a unit of angular measurement used in mathematics to measure angles in terms of the radius of the circle. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. This makes the full circle total \( 2\pi \) radians, since the circumference of a circle is \( 2\pi r \) where \( r \) is the radius. Hence, by converting degrees to radians, one can find that \( 360^\circ \) equals \( 2\pi \) radians.

The beauty of radians lies in their natural connection to the arc and the radius, making them an intrinsic part of circular motion and periodic phenomena. Using radians allows for direct linking of angular displacement to linear displacement along the circle's circumference and serves as the standard unit of angular measure in the field of mathematics, which simplifies many mathematical expressions.

Angular Measurement

Angular measurement is a way of describing the size of an angle or the amount of rotation. It is crucial in understanding the rotation of a point around a fixed origin in Cartesian coordinates, particularly when converting to polar coordinates. In the context of polar coordinates, angular measurement provides a way to indicate direction from a fixed reference point. This measurement can be express in degrees or radians, with one complete revolution around a circle being equivalent to \( 360^\circ \) or \( 2\pi \) radians.

When it comes to the polar coordinate system, it's essential to realize that adding a full rotation of \( 2\pi \) radians to the angle \( \theta \) doesn't change the point's location; it only brings the point back to its original position. This property underlies the concept that the coordinates \( (r, \theta) \) and \( (r, \theta+2 \pi) \) represent the same point. It exemplifies how the polar coordinate system can express a single location with multiple angle values, differing by multiples of full rotations. This unique characteristic is a direct consequence of the angular measurement's circular nature in the polar context.