Problem 4 - Contemporary Precalculus

Question

Plot the point whose polar coordinates are given. $$(-2,2 \pi / 3)$$

Step-by-Step Solution

Answer

Question: Plot the point $$(-2, 2\pi/3)$$ in the Cartesian plane, given in polar coordinates. Answer: The Cartesian coordinates of the point are $$(1, -\sqrt{3})$$. To plot this point, start at the origin, move 1 unit to the right and then 1.73 units down.

Step 1: Analyze the Given Polar Coordinates

We're given polar coordinates $$(-2, 2\pi/3)$$, where \(r = -2\) and \(\theta = 2\pi/3\). We need to convert these coordinates to Cartesian coordinates to plot them.

Step 2: Apply Conversion Formulas

Using the conversion formulas, calculate the Cartesian coordinates of the point: $$x = r\cos(\theta) = (-2)\cos(2\pi/3)$$ $$y = r\sin(\theta) = (-2)\sin(2\pi/3)$$

Step 3: Calculate the Coordiantes

Calculate the values of \(x\) and \(y\): $$x = (-2)\cos(2\pi/3) = (-2)\left(-\frac{1}{2}\right) = 1$$ $$y = (-2)\sin(2\pi/3) = (-2)\left(\frac{\sqrt{3}}{2}\right) = -\sqrt{3}$$ So the Cartesian coordinates are \((1, -\sqrt{3})\).

Step 4: Plot the Point on the Graph

Finally, plot the point \((1, -\sqrt{3})\) on the Cartesian plane. Start at the origin, move 1 unit to the right (since \(x = 1\)), and then move down approximately 1.73 units (since \(y = -\sqrt{3}\)). Mark the point and label it as \((1, -\sqrt{3})\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates, also known as rectangular coordinates, provide a method to identify the location of a point in a two-dimensional space using two numbers, usually denoted as \( (x, y) \). These numbers correspond to the point's horizontal and vertical distances from a fixed reference point called the origin, denoted as \( (0, 0) \). The \( x \) value is known as the abscissa and represents the point's distance to the right or to the left of the origin, while the \( y \) value, called the ordinate, represents the point's distance above or below the origin.

A Cartesian coordinate system is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points in the upper right area of the plane have positive \( x \) and \( y \), those in the upper left have negative \( x \) and positive \( y \) values, and so on for the lower quadrants. This system is incredibly useful for graphing equations, modeling real-world situations, and carrying out various mathematical and engineering calculations.

Coordinate Conversion

Coordinate conversion is the process of transitioning between different coordinate systems, such as converting polar coordinates to Cartesian coordinates or vice versa. Polar coordinates denote a point in a plane using a radius and an angle relative to the positive x-axis. Unlike Cartesian coordinates, which use a grid, polar coordinates rely on a circle with the same origin.

To convert from polar to Cartesian coordinates, we use the formulas:\
\begin{align*}) x &= r \cos(\theta)\,\end{align*}\
\begin{align*}) y &= r \sin(\theta)\,\end{align*}\
Where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle in radians measured counterclockwise from the positive x-axis. To convert from Cartesian to polar, the formulas are:\
\begin{align*}) r &= \sqrt{x^2 + y^2}\,\end{align*}\
\begin{align*}) \theta &= \arctan \left(\frac{y}{x}\right)\,\end{align*}\
if \( x > 0 \), with modifications needed depending on the quadrant in which the \( (x, y) \) coordinates lie. This practice is especially relevant in fields such as physics, where polar coordinates are often more natural for describing circular motion or orbital paths.

Plotting Points

Plotting points is a fundamental skill in graphing, allowing one to visually represent a set of values on a coordinate system. Whether you're plotting on Cartesian coordinates or another system, the method involves identifying and marking a precise location based on given numerical values or equations.

When plotting points in Cartesian coordinates, start at the origin (0,0). To plot the point \( (1, -\sqrt{3}) \), first move 1 unit to the right on the x-axis from the origin, since the x-coordinate is positive. Then, go down (because the y-coordinate is negative) roughly 1.73 units, which is the decimal equivalent of \( -\sqrt{3} \). Place a dot or other marker on the coordinate plane to represent the point and label it with its coordinates for clarity.

Plotting is often done with pencil and graph paper, but with advances in technology, various software and online platforms can plot points with greater precision and less effort. It remains an invaluable tool in visualizing and solving mathematical problems, in contexts ranging from simple homework exercises to advanced scientific research.