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 \(r>0\) . $$ (-1,7 \pi / 6) $$

Step-by-Step Solution

Answer

Point (-1, 7π/6) equals (1, 13π/6) and stays (-1, 7π/6) with r < 0.

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 from the positive x-axis. In this exercise, we start with \((-1, 7\pi/6)\).

Step 2: Plotting the Original Point

To plot the point \((-1, 7\pi/6)\), note that \(-r\) indicates moving in the opposite direction. Solve by plotting \((1, 7\pi/6 + \pi)\), because \((r, \theta)\) becomes \((-r, \theta+\pi)\) when \(r<0\).

Step 3: First Alternative Representation (r < 0)

For an alternative polar coordinate with \(r<0\), keep \((-1, 7\pi/6)\) as it already satisfies the condition that \(r<0\).

Step 4: Second Alternative Representation (r > 0)

Convert \((-1, 7\pi/6)\) to use a positive \(r\) by using \(r > 0\). Adjust the angle by adding \(\pi\): \((1, 7\pi/6 + \pi) = (1, 13\pi/6)\) as this points in the same physical location.

Key Concepts

Radial Distance

In the world of polar coordinates, the term 'radial distance' plays a crucial role. It is the distance from the origin, also known as the pole, to the point in the plane. This distance is represented by the symbol \( r \) in polar coordinates, \((r, \theta)\). To visualize radial distance, imagine plotting a point using a set of concentric circles that expand outward from a central point (the origin). Note that when the radial distance \( r \) is positive, the point falls on your specified angle directly. Conversely, for a negative \( r \), the point appears in the direction opposite to the specified angle. When you interpret radial distance in problems, it's important to recognize how this impacts the location of points. For example, the point \((-1, 7\pi/6)\) initially suggests moving \(-1\) unit from the origin, but in the opposite direction of the angle \(\theta\). This concept becomes useful when converting or analyzing different representations of the same point.

Angle from the Positive X-Axis

Angles in polar coordinates, denoted by \( \theta \), show how far off a point is from the positive x-axis. This angle is measured in standard position, meaning it starts at the positive x-axis and moves counter-clockwise. Understanding this angle is essential because any angle in polar coordinates helps to pinpoint the direction of the radial distance.In our exercise, the angle \( 7\pi/6 \) radians is crucial. In standard terms, it references the direction of \( 210^\circ \) on the Cartesian plane. Visualizing this, it means moving past the positive x-axis (0 radians) and wrapping around slightly beyond \( \pi \) or \( 180^\circ \). This reference angle aids in plotting the point by indicating the trajectory along which the radial distance extends, whether positive or negative. Thus, recognizing and working with angles is a fundamental skill in using polar coordinates, leading to understanding complex plots and conversions.

Alternative Polar Representation

One unique aspect of polar coordinates is that a single point can have multiple representations. This is because angles can be adjusted by adding \( 2\pi \) radians (or full rotations), and the radial distance can shift between positive and negative forms, depending on the approach.Consider the original point \((-1, 7\pi/6)\). In polar coordinates, this has a negative radial distance and an angle in one direction. To find different representations:

• For an alternative with \( r < 0 \), keep the original representation since \( r = -1 \) already satisfies the condition.
• For an alternative with \( r > 0 \), adjust the coordinates such that the radial distance is positive. This involves converting \((-1, 7\pi/6)\) to \((1, 13\pi/6)\), achieved by moving \( r \) to positive \( 1 \) and effectively adding \( \pi \) to the angle \( \theta \) to transition direction.

Thus, understanding the flexibility in polar coordinates helps adjust angles and radial distances, emphasizing the richness and versatility of this coordinate system.