Problem 7 - 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 \(r>0\). $$ (3, \pi / 2) $$

Step-by-Step Solution

Answer

The original coordinates are \((3, \pi/2)\); other representations are \((-3, 3\pi/2)\) and \((3, -3\pi/2)\).

Step 1: Understanding Polar Coordinates

Polar coordinates are in the form \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis counter-clockwise.

Step 2: Plot the Given Point

To plot \((3, \pi/2)\), start from the origin, move straight up along the y-axis since the angle \(\pi/2\) is 90 degrees. Measure a distance of 3 units from the origin along this path.

Step 3: Finding Representation with Negative Radius

For a representation with \(r < 0\), reflect the existing position across the origin. The negative radius will be \(-3\). To find the new angle, add \(\pi\) to the original angle: \(\pi/2 + \pi = 3\pi/2\). Thus, one representation is \((-3, 3\pi/2)\).

Step 4: Finding Another Positive Radius Representation

To find another representation with \(r > 0\), alter the angle by adding or subtracting full rotations \(2\pi\). Subtract \(2\pi\) from the original angle: \(\pi/2 - 2\pi = -3\pi/2\). This makes the coordinates \((3, -3\pi/2)\).

Key Concepts

Angle Measurement

In polar coordinates, the angle \(\theta\) is a crucial part of locating a point. It tells us how much to rotate around the origin from the positive x-axis to reach our desired direction.
This angle is typically measured in radians, which is a standard unit of angular measure used in many areas of mathematics. One full circle is \(2\pi\) radians, equivalent to 360 degrees.When working with angles, remember:

• \(\pi/2\) radians corresponds to 90 degrees, pointing straight up along the positive y-axis.
• Positive angles are usually measured counter-clockwise from the positive x-axis.
• Negative angles, although less common in initial plots, simply mean clockwise rotation.

Understanding angle measurement helps plot points correctly and provides clarity on how angles affect direction.

Positive and Negative Radius

The radius \(r\) in polar coordinates indicates the distance from the origin to the point. The concept of positive and negative radius is essential when representing the same point in different ways. A positive radius ( \(r>0\) ) means the point is located at that distance in the direction of \(\theta\). For instance, a point like \(3\, \pi/2\) is 3 units away from the origin straight up.
A negative radius ( \(r<0\) ) means we are moving in the opposite direction of \(\theta\). To represent it:

• Reflect the point across the origin.
• Add \(\pi\) (180 degrees) to the angle to adjust direction.

The resulting pair, such as \((-3, 3\pi/2)\), will still point to the originally plotted location, ensuring accuracy in multiple representations.

Coordinate Representation

Polar coordinates offer flexibility in representing the same point in multiple ways. This versatility extends from the basic use of positive and negative radii to the adjustment of angles. A single polar point can be represented with an infinite number of equivalent coordinates by taking advantage of the full rotation.
For example:

• Adding or subtracting \(2\pi\) (a full circle) to the angle \(\theta\) doesn't change the point's location. For instance, \((3, \pi/2)\) can also be written as \((3, -3\pi/2)\) by using the periodic nature of angles.
• Changing the sign of the radius while rotating the angle by \(\pi\) (such as \((-3, 3\pi/2)\)) achieves the same point.

This adaptability is invaluable when solving problems, allowing us to choose the most convenient representation for the task at hand.