Problem 70 - Precalculus

Question

Use a graphing utility to find one set of polar coordinates of the point given in rectangular coordinates. $$\left(-\frac{7}{9},-\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

Please use a graphing utility to correctly find the polar coordinates. The process has been explained in the steps, but the exact answer depends on precision of the graphing utility used.

Step 1: Understand the polar coordinates

Polar coordinates are usually represented in the form \((r, \theta)\), where \(r\) is the radial coordinate (distance from the origin) and \(\theta\) is the angular coordinate (angle from the positive x-axis).

Step 2: Plot the point in rectangular coordinates

For the given rectangular coordinates \(\left(-\frac{7}{9},-\frac{3}{4}\right)\), plot this point on the Cartesian plane using a graphing utility. This can be done by plotting the x-coordinate (-7/9) on the x-axis and the y-coordinate (-3/4) on the y-axis.

Step 3: Convert rectangular coordinates to polar coordinates

After the point is plotted, we can determine the polar coordinates. To do this, calculate the length of line between the origin and the point to find 'r', and measure the angle the line forms with positive x-axis in counter clockwise direction for \(\theta\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of representing points in a two-dimensional plane using two numbers, typically \(x\) and \(y\). These coordinates tell us the horizontal and vertical positions of a point relative to an origin point (0,0). In essence, rectangular coordinates give us an easily understandable grid system to locate points.

For example, a point described by the coordinates \((-\frac{7}{9}, -\frac{3}{4})\) means it is located \-\frac{7}{9}\ units to the left, and \-\frac{3}{4}\ units down from the origin. This negative sign in the coordinates indicates movement in the opposite direction along both the x and y axes. Using these coordinates is useful for graph plotting and analyzing geometric shapes.

Graphing Utility

A graphing utility is a tool, often software or a calculator, that helps visualize mathematical concepts by plotting them on a graph. These tools are invaluable when you need to understand complex data by transforming abstract numbers into easily interpretable graphical representations.

For our coordinate conversion exercise, the graphing utility aids in accurately plotting the initial rectangular coordinates onto a Cartesian plane. This ensures the conversion to polar coordinates is based on precise visual information.

• Accurate plotting helps visualize the position, making it easier to calculate the distance from the origin.
• The angle relative to the positive x-axis becomes clearer when plotted.

This step is crucial to ensure proper understanding and conversion of coordinates.

Coordinate Conversion

Coordinate conversion is the process of translating information from one type of coordinate system to another. Commonly, in mathematics, this involves converting rectangular coordinates to polar coordinates or vice versa.

For converting rectangular coordinates like \((-\frac{7}{9}, -\frac{3}{4})\) into polar coordinates \( (r, \theta) \), you need to:

• Calculate the radial distance \(r\), using the formula \( r = \sqrt{x^2 + y^2} \).
• Find the angle \(\theta\) by using the arctangent function, \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

Keep in mind that \(\theta\) is measured in radians in mathematics. Adjusting \(\theta\) based on the quadrant where the point lies ensures an accurate conversion. This conversion is essential in various fields such as physics and engineering, where different tasks might require specific coordinate systems.

Radial Coordinate

The radial coordinate, represented oftentimes by \(r\) in polar coordinates, denotes the distance from a given point to the origin on the graph. Unlike the straightforward x and y coordinates in the Cartesian system, \(r\) represents a magnitude, which is always a positive value irrespective of the point's position on the graph.

To calculate the radial coordinate of a point given in rectangular format, the formula \( r = \sqrt{x^2 + y^2} \) is employed. For the specific example of \((-\frac{7}{9}, -\frac{3}{4})\), you plug these values into the formula:

• \[ r = \sqrt{\left(-\frac{7}{9}\right)^2 + \left(-\frac{3}{4}\right)^2} \]
• This will give the straight-line distance from the origin to the point, forming one half of the polar coordinate pair.

Remember, the radial coordinate is crucial in understanding the geometric relation of points relative to the origin, especially in scenarios involving rotational dynamics.