Problem 11 - 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

The polar coordinates \((5, \pi)\) and \((-5, 2\pi)\) both represent the same point.

Step 1: Understanding Polar Coordinates

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The given coordinates are \((-5, 0)\). Thus, \(r = -5\) and \(\theta = 0\). This means the point is 5 units away from the origin on the negative x-axis.

Step 2: Plotting the Point

To plot \((-5, 0)\), we first locate \(\theta = 0\), which aligns with the positive x-axis. Since \(r = -5\), we move 5 units in the opposite direction of the angle, placing the point at \((-5, 0)\) on the Cartesian coordinate system.

Step 3: Finding an Equivalent Polar Coordinate with \(r > 0\)

To find a polar coordinate with \(r > 0\), we must add \(\pi\) radians to the angle to reverse the direction, thus \(r = 5\) and \(\theta = 0 + \pi = \pi\). Therefore, \((5, \pi)\) is an equivalent representation of the point.

Step 4: Finding an Equivalent Polar Coordinate with \(r<0\)

The original point already has \(r = -5\), so it fits the criteria of \(r < 0\). However, to find another representation, add \(2\pi\) to the angle, giving \(\theta = 0 + 2\pi = 2\pi\). Thus, another equivalent point is \((-5, 2\pi)\).

Key Concepts

Plotting Points

Plotting points using polar coordinates involves a unique method of representation compared to the more familiar Cartesian system. Here, each point is defined by two values: a radius and an angle. This pair is noted as \((r, \theta)\).

• Radius (\(r\)): It indicates how far the point is from the origin (or center of the polar grid).
• Angle (\(\theta\)): It tells us the direction, measured in radians, from the positive x-axis.

When plotting the point \((-5, 0)\):
1. Start by locating the angle \(\theta = 0\), which lies on the positive x-axis.
2. Because the radius is negative at \(-5\), move 5 units in the opposite (negative) direction along the x-axis.
This step-by-step approach makes it easier to accurately place points on a graph using polar coordinates, especially when the radius or angle varies from the standard forms.

Negative Radius

In polar coordinates, the radius can indeed be negative, which might seem unusual at first. A negative radius essentially means you are plotting the point by moving the specified distance in the opposite direction of the given angle.
When we talk about \((-5, 0)\):

• Opposite Direction: Normally, you would move 5 units along the angle of 0, or the positive x-axis, but the negative sign instructs us to instead move backwards, placing the point on the negative x-axis.
• Point Location: The result is the same spatial position as moving +5 units at an angle of \(\pi\) (180 degrees), showing that polar coordinates allow for flexible representations.

This trait of polar coordinates provides a more robust system of plotting, which can represent a wide array of points while offering multiple valid coordinate sets for the same location.

Angle Conversion

Converting angles is an essential aspect when working with polar coordinates because it allows for different but equivalent representations of a point. Multiply by different multiples of \(\pi\) or \(2\pi\) to change the direction without altering the point's position. For instance:

• For a positive radius: Starting at \(\theta = 0\) with \(r=5\), if you convert it by adding \(\pi\), the angle becomes \(\theta = \pi\), thus flipping direction, and you reach the same point.
• For a negative radius: Initially, the negative radius \(r=-5\) at \(\theta = 0\) can use the same technique to adjust the angle. Adding \(2\pi\), so \(\theta = 2\pi\), keeps the point in the same location as it simply represents a full circle.

This method of conversion ensures that students can intuitively adapt coordinates to favorable formats for calculation and graphing. Remember, any multiple of \(2\pi\) added to or subtracted from \(\theta\) will point the radius to an equivalent angle on the polar grid.