Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(0,-\frac{7 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates corresponding to the polar coordinates \((0, -\frac{7 \pi}{6})\) is \((0, 0)\)

Step 1: Understand polar coordinates

First, polar coordinates \(r, \theta\) represent a point in the plane as a distance \(r\) from the origin and an angle \(\theta\) in radians, measured counter-clockwise from the positive x-axis. Here, we are given the polar coordinates \(0, -\frac{7 \pi}{6}\). This means that the point is origin, because the radius \(r = 0\).

Step 2: Convert polar coordinates to rectangular coordinates

To convert these to rectangular coordinates \((x, y)\), we normally use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). But in our case, as \(r = 0\), applying the formulas directly will give \(x = 0\) and \(y = 0\).

Step 3: Plot the point

Plot this point (0,0) on the Cartesian plane. This is the origin.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, help us pinpoint a location on a flat plane. Imagine drawing two intersecting lines: the horizontal is the x-axis, and the vertical is the y-axis. Together, they create a grid that maps locations using pairs of numbers in the form \((x, y)\).

• X-coordinate: Represents the horizontal position.
• Y-coordinate: Represents the vertical position.

Each point on this grid corresponds to a unique pair of values. For example, the point \((2, 3)\) is 2 units along the x-axis and 3 units up the y-axis. This concept is fundamental for understanding how positions are described in a two-dimensional space using numbers.

Coordinate Conversion

Coordinate conversion helps us translate between polar and rectangular coordinates. Polar coordinates, defined by \((r, \theta)\), describe a position through a distance from the origin and an angle from the positive x-axis. To convert these to rectangular coordinates \((x, y)\), we use:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

In the given problem, because \(r = 0\), this means we have no distance from the origin and the point remains at \((0,0)\), regardless of the angle \(-\frac{7 \pi}{6}\). This conversion process is crucial for graphing and analysis, bridging the gap between circular and grid-based systems.

Graphing Points

Graphing points is a simple yet powerful skill for visualizing data. When we graph a point on a Cartesian plane, it shows exactly where that point resides in the space defined by the axes. Let's break down the process:

• Identify the coordinates \((x, y)\).
• Locate the x-value on the horizontal axis.
• Locate the y-value on the vertical axis.
• Draw a dot where these two values meet.

In our example, the point \((0,0)\) represents the origin. It's the central point where both axes meet. Understanding how to graph these points helps in many areas of math and science, giving you a clear picture of relationships and structures in data.