Problem 9 - Precalculus Mathematics for Calculus

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

Plot the point at 3 units up along the positive y-axis; other representations are \((-3, 3\pi / 2)\) and \((3, 5\pi / 2)\).

Step 1: Understanding Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis, counterclockwise. In this exercise, we are given \((3, \pi / 2)\), which means a distance of 3 units from the origin at an angle of \(\pi / 2\) radians.

Step 2: Plot the Initial Point

To plot \((3, \pi / 2)\), start from the origin and move along the angle \(\pi / 2\). In the standard coordinate plane, \(\pi / 2\) is oriented along the positive y-axis. Move 3 units in this direction. The point will be plotted 3 units above the origin, on the y-axis.

Step 3: Find Representation with r < 0

To find a representation with \(r < 0\), we need to invert the direction and adjust the angle by adding \(\pi\) (180 degrees). Starting with \(3, \pi / 2\), invert the distance to \(-3\) and adjust the angle: \(\theta = \pi / 2 + \pi = 3\pi / 2\). Hence, an equivalent representation with \(r < 0\) is \((-3, 3\pi / 2)\).

Step 4: Find Representation with r > 0

The given coordinates \((3, \pi / 2)\) already have \(r > 0\). However, another representation can be found by adding \(2\pi\) to \(\theta\): \(\theta = \pi / 2 + 2\pi = 5\pi / 2\). So, another equivalent coordinate is \((3, 5\pi / 2)\).

Key Concepts

Plotting Points in Polar Coordinates

In polar coordinates, points are represented using a distance and an angle. The distance, noted as \(r\), indicates how far the point is from the origin, akin to how coordinates work in a radial fashion rather than the usual grid system. The angle, \(\theta\), determines the direction of the point from the origin, measured counterclockwise from the positive x-axis.
When it comes to plotting a point, consider the coordinates \((3, \pi/2)\). Here, you start by measuring the angle \(\pi/2\) radians from the x-axis. Visualize \(\pi/2\) as a 90° rotation, aligning with the positive y-axis. Proceed to measure the distance \(r\) of 3 units from the origin along this direction. As a result, you'd plot the point at \(3\) units on the y-axis.

Coordinate Conversion in Polar Form

Finding equivalent polar coordinates requires understanding how to manipulate \(r\) and \(\theta\). Changing \(r\) to negative doesn't move the point; instead, it flips its direction through the origin. If adjusting the angle isn't enough, consider adding or subtracting multiples of \(2\pi\) to \(\theta\), looping the radial path.
For instance, to represent the point with \(r < 0\), reverse the original \(r=3\) to \(-3\). By adding \(\pi\) to the angle \(\theta = \pi/2\), the new angle becomes \(3\pi/2\). So, the coordinates become \((-3, 3\pi/2)\).
Alternatively, a new \(r > 0\) form is possible. From \(\theta = \pi/2\), simply add \(2\pi\) to create an equivalent \(\theta = 5\pi/2\). The positive form \((3, 5\pi/2)\) now defines your point again.

Angle Measurement in Polar Coordinates

Angles in polar coordinates are always measured starting from the positive x-axis, heading counterclockwise. They can either be in degrees or radians, but radians are commonly used in mathematics due to their natural relation to the unit circle.
Understanding angle equivalence is crucial since angles greater than \(2\pi\) or less than zero essentially map back into the initial circle range, creating repeated paths.
For the exercise, starting angle \(\theta = \pi/2\) aligns with the positive y-axis. If you imagine adding \(2\pi\) to it, you would traverse a full circle, arriving again in the same direction. Hence, \(5\pi/2\) and \(\pi/2\) dictate the same path, showing the coherence of angle measurement in plotting and conversions.