Problem 12 - Precalculus with Calculus Previews

Question

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (3,7 \pi / 6) $$

Step-by-Step Solution

Answer

(a) \((3, -5\pi/6)\), (b) \((3, 19\pi/6)\), (c) \((-3, 13\pi/6)\), (d) \((-3, -11\pi/6)\)."

Step 1: Identify Original Coordinates

The given point in polar coordinates is \((r, \theta) = (3, 7 \pi / 6)\). This means that the radius \(r = 3\) and the angle \(\theta = 7 \pi / 6\).

Step 2: Convert Angle to Negative (a)

To find an alternative polar coordinate with \(r>0\) and \(\theta<0\), subtract \(2 \pi\) from the angle: \(\theta = 7 \pi / 6 - 2 \pi = 7 \pi / 6 - 12 \pi / 6 = -5 \pi / 6\). So the new coordinates are \((3, -5 \pi / 6)\).

Step 3: Convert Angle Over 2π (b)

To find a coordinate with \(r>0\) and \(\theta>2 \pi\), add \(2 \pi\) to the angle: \(\theta = 7 \pi / 6 + 2 \pi = 7 \pi / 6 + 12 \pi / 6 = 19 \pi / 6\). So the new coordinates are \((3, 19 \pi / 6)\).

Step 4: Find Negative Radius Coordinate (c)

For \(r<0\) and \(\theta>0\), flip the radius and add \(\pi\) to the angle since coordinates are periodic: \(r = -3\) and \(\theta = 7 \pi / 6 + \pi = 13 \pi / 6\). The new coordinates are \((-3, 13 \pi / 6)\).

Step 5: Find Negative Radius and Negative Angle (d)

To find \(r<0\) and \(\theta<0\), keep \(r = -3\) and find the negative equivalent of \(13 \pi / 6\) by subtracting \(2 \pi\): \(\theta = 13 \pi / 6 - 2 \pi = 13 \pi / 6 - 12 \pi / 6 = \pi / 6\); the angle should be expressed negatively, so subtract \(2\pi\) again to get \(-11 \pi / 6\). Hence, the coordinates are \((-3, -11 \pi / 6)\).

Key Concepts

Alternative Coordinates

In polar coordinates, a single point can have multiple representations. This is due to the periodic nature of angles and the freedom to choose a positive or negative radius. The representation of the point can change according to the specific conditions you want, such as having a positive or negative angle, or changing the direction with a negative radius.
By understanding how to alter the radius and the angle while maintaining the point's identity, you can find various representations that still point to the same location on the plane. The essence lies in manipulating these dimensions properly based on the given conditions.

Negative Angles

When we talk about negative angles in terms of polar coordinates, we mean angles that are measured clockwise from the positive x-axis. Since polar angles are typically measured in a counter-clockwise direction, negative angles are less than zero.
To convert a positive angle to a negative equivalent, we subtract multiples of full rotations, which are represented by increments of \(2\pi\), until the angle is negative. For example, an angle like \(7 \pi / 6\) can be re-expressed in negative terms by subtracting \(2\pi\), arriving at \(-5 \pi / 6\). This still points to the same spacial direction as the original angle.

Positive Angles

In polar coordinates, positive angles are those measured counter-clockwise from the positive x-axis. Normally, these angles range between \(0\) and \(2\pi\). However, they can exceed \(2\pi\), making full rotational cycles and beyond possible.
For example, to convert \(7 \pi / 6\) to an angle greater than \(2\pi\), simply add \(2\pi\) resulting in \(19 \pi / 6\). This transformation allows for a different numeric representation of the angle, while pointing in the same spatial direction.

Negative Radius

A negative radius in polar coordinates essentially reflects the point across the origin. This means the point is moved to the opposite side of its original location, along the same line determined by the angle.
To find an equivalent negative radius coordinate, we flip the sign of the radius and add \(\pi\) radians to the angle to maintain the point's direction, incorporating another half rotation. For example, if \(r = 3\) at \(7 \pi / 6\), then \(r = -3\) will be at \(13 \pi / 6\), ensuring the point lies on the same line extended in the opposite direction.