Problem 13 - Algebra and Trigonometry

Question

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (-5,0) $$

Step-by-Step Solution

Answer

Plot as (-5,0). Alternates: (5,π) and (-5,2π).

Step 1: Understand Polar Coordinates

Polar coordinates represent a point in the plane using a radius \(r\) and an angle \(\theta\). The coordinates \((-5,0)\) indicate a point 5 units away from the origin in the direction opposite to the positive x-axis (since angle \(0\) corresponds to the positive x-axis and \(r = -5\) means go in the opposite direction).

Step 2: Plot the Given Point

To plot \((-5, 0)\), start from the origin and move 5 units to the left along the x-axis. The point will lie on the negative x-axis, 5 units away from the origin.

Step 3: Find a Polar Representation with \(r > 0\)

To find an equivalent point with \(r > 0\), we can use the angle \(\theta = \pi\) radians. Therefore, the coordinates are \((5, \pi)\), where \(r = 5\) is positive, and the direction \(\pi\) radians places the point on the negative x-axis.

Step 4: Find a Polar Representation with \(r < 0\)

Given the point is already with \(r < 0\), another equivalent representation can be obtained by going full circle plus the original angle: \((-5, 2\pi)\). This keeps the radius negative while completing 360 degrees (or \(2\pi\) radians) from the original direction.

Key Concepts

Radius

In polar coordinates, the radius, denoted as \(r\), describes how far a point is from the origin.
It works similarly to a distance in Cartesian coordinates, but rather than being a distance from the origin along a straight line, it is measured as a radial distance from the pole (the origin in polar systems).
This radius is always a scalar quantity and can be either positive or negative, indicating direction relative to the angle.

• A positive radius means the point lies in the direction of the angle \(\theta\).
• A negative radius implies the point is in the directly opposite direction of \(\theta\).

Understanding the sign of \(r\) is key to determining the correct position of the point on the plane.

Angle

The angle \(\theta\) in polar coordinates specifies the direction of the point relative to the positive x-axis. This angle is measured in either degrees or radians and is a central aspect of mapping any point in a polar coordinate system.
The angle \(\theta\) starts from the positive x-axis and moves counter-clockwise.

• An angle of \(0\) radians or \(0\) degrees aligns directly with the positive x-axis.
• An angle of \(\pi\) radians or \(180\) degrees points directly opposite.
• It's crucial to understand that angles can be greater than \(2\pi\) radians, indicating a full circle plus additional rotation.

Utilizing angles allows polar equations to offer symmetric insights not available in Cartesian systems.

Coordinate Representation

Polar coordinate representation describes points using a radius and an angle: \((r, \theta)\). Each pair of a radius and angle corresponds to a unique point, but points can have multiple representations.
This multiple representation nature comes from:

• The ability to add or subtract \(2\pi\) from \(\theta\), achieving equivalent points.
• Using positive or negative values for \(r\).

For instance, the point represented by \((-5, 0)\) is equivalent to \((5, \pi)\) or \((-5, 2\pi)\). Discovering these equivalent representations allows flexibility in analyzing and simplifying problems in polar form.

Negative Radius

A negative radius \(r\) indicates an opposite direction to the angle \(\theta\). When you plot a point with a negative radius, instead of moving in the direction the angle points, you head in the exact opposite direction.
For example, with coordinates \((-5, 0)\), rather than heading right along the x-axis for a positive \(r\), you would move left because of the negative value.

• This indicates the same location as a positively radiused point at an angle shifted by \(\pi\) radians.

Understanding this concept allows conversion between equivalent polar coordinate forms and facilitates transitions between positive and negative radius descriptions.

Positive Radius

A positive radius \(r\) provides a straightforward interpretation—moving from the origin in the direct path of the angle \(\theta\). For a given point \((r, \theta)\), if \(r\) is positive, the point is located \(r\) units away from the origin in the direction that \(\theta\) specifies.
In our solution, we adapted the coordinate \((-5, 0)\) to a positive radius form: \((5, \pi)\). Here:

• The positive \(r=5\) moves 5 units outward from the origin.
• The angle \(\pi\) directs this movement from the positive x-axis to the negative side.

Recognizing how radius direction influences representation is crucial for navigating and converting polar coordinates efficiently.