Problem 14 - Essential Calculus

Question

The rectangular coordinates of a point are given. Plot the point and find \(t w o\) sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (3,-\sqrt{3}) $$

Step-by-Step Solution

Answer

The two sets of polar coordinates for the point \(3,-\sqrt{3}\) are \((2\sqrt{3},\frac{5\pi}{6})\) and \((2\sqrt{3},\frac{-π}{6})\) or equivalently \((2\sqrt{3},\frac{11\pi}{6})\).

Step 1: Plotting the point

We plot the given point (3,-√3) in a Cartesian coordinate system. It will be in the fourth quadrant, as the x-coordinate is positive and the y-coordinate is negative.

Step 2: Calculate the radius

The radius \(r\) for polar coordinates is given by \[r=\sqrt{x^2+y^2}\]. In this case, \(r=\sqrt{3^2+(-\sqrt{3})^2}=2\sqrt{3}\].

Step 3: Calculate the angle θ

The angle for polar coordinates is typically measured counterclockwise from the positive x-axis. We calculate θ by \[θ=\arctan\left(\frac{y}{x}\right)\] or \[θ=\arctan\left(\frac{-\sqrt{3}}{3}\right)\]. θ will be a negative angle as the point is in the fourth quadrant.

Step 4: Convert the angle to positive and in the range of 0 to 2π

We add 2π to the negative θ to get it back into the required range. This gives us two sets of polar coordinates for the original point. First when we are evaluating for θ in 0 to 2π counterclockwise in which θ will be an angle in the first quadrant. The second will be when θ measures its angle 2π to 0 clockwise which will plot θ in the fourth quadrant.

Key Concepts

Rectangular Coordinates

Rectangular coordinates help us pinpoint a location on a two-dimensional plane using a pair of numbers. Typically, a point is represented as \(x, y\). In this example, the coordinates are \(3, -\sqrt{3}\). The first number, \(3\), tells us how far to move horizontally from the origin, while the second number, \(-\sqrt{3}\), indicates the vertical movement. When both coordinates are combined, they allow us to plot the exact position of the point on the plane.
Using these coordinates, you can visualize the point as a specific spot on a grid, much like how you might locate a position on a map.

Cartesian Coordinate System

The Cartesian coordinate system is the framework we use to understand and plot rectangular coordinates. Imagine a grid made up of two intersecting lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, each serving as a reference for plotting points.

• The first quadrant is where both x and y are positive.
• The second has a negative x and a positive y.
• The third quadrant has both negative coordinates.
• The fourth shows a positive x and a negative y, where our point \(3, -\sqrt{3}\) is located.

Using this system helps convert between different types of coordinate systems.

Radius Calculation

The radius in polar coordinates represents the distance from the origin to the point. To calculate it, we use the Pythagorean theorem formula: \(r = \sqrt{x^2 + y^2}\). This formula takes into account both the horizontal and vertical distances specified by the rectangular coordinates.
For our example, substituting the values gives:
\[ r = \sqrt{3^2 + (-\sqrt{3})^2} = 2\sqrt{3} \]
This calculation tells us the length of the line connecting the point to the origin, forming a radius useful for polar coordinates.

Angle Calculation

The angle is measured from the positive x-axis to the line connecting the origin and our point. This angle, \(\theta\), helps us understand the direction of the point. We calculate it using the formula:
\[ \theta = \arctan\left(\frac{y}{x}\right) \]
For the point \(3, -\sqrt{3}\), the calculation becomes:
\[ \theta = \arctan\left(\frac{-\sqrt{3}}{3}\right) \]
Depending on the quadrant, we adjust this angle to fit within \[0, 2\pi)\]. In this case, since the angle is initially negative, we add \2\pi\ to bring it back into the positive range. By fully understanding the angle, we can express the point in polar coordinates accurately.