Problem 141 - Calculus Volume 3

Question

The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2?]. Round to three decimal places. $$ (3,-\sqrt{3}) $$

Step-by-Step Solution

Answer

The polar coordinates are approximately \((3.464, \frac{11\pi}{6})\) and \((3.464, -\frac{\pi}{6} + 2\pi)\).

Step 1: Understand the Rectangular Coordinates

You are given the coordinates \((3, -\sqrt{3})\). These are in Cartesian (rectangular) form where \(x = 3\) and \(y = -\sqrt{3}\). Our goal is to convert them into polar coordinates \((r, \theta)\) such that \(0 < \theta \leq 2\pi\).

Step 2: Calculate the Radius (r)

The radius \(r\) in polar coordinates is calculated using the formula \(r = \sqrt{x^2 + y^2}\). So, substitute the values:\[r = \sqrt{3^2 + (-\sqrt{3})^2} = \sqrt{9 + 3} = \sqrt{12} = 2\sqrt{3}\approx 3.464\]Thus, \(r \approx 3.464\).

Step 3: Find the Angle (θ) for the Principal Value

The angle \(\theta\) can be found using \(\tan\theta = \frac{y}{x}\). Therefore, \(\tan\theta = \frac{-\sqrt{3}}{3}\). Solving \(\tan\theta = -\frac{1}{\sqrt{3}}\), we know \(\theta = -\frac{\pi}{6}\) in the fourth quadrant of the unit circle. Adjusting for the range, \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\approx 5.760\).

Step 4: Determine the Second Angle θ for the Polar Coordinates

The second representation of the angle is found by adding \(2\pi\) (a full rotation) to the principal angle. Thus, the second angle \(\theta = \frac{11\pi}{6} - 2\pi = -\frac{\pi}{6}\implies\theta = \frac{11\pi}{6}\) is the only angle that lies within \((0, 2\pi]\) as this is the same angle after correction.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a two-dimensional plane using an ordered pair of numbers. These numbers correspond to the distance from each of the two perpendicular axes, typically labeled as the x-axis and y-axis. When you have a coordinate pair like

• \((3, -\sqrt{3})\),

this tells you the point is located 3 units along the positive direction of the x-axis and \(-\sqrt{3}\) units along the negative direction of the y-axis.
Rectangular coordinates are often used for their simplicity and ease in calculations concerning linear and vector equations. However, they are not always the most convenient for dealing with circular or rotational problems, which leads us to the need for conversion to other forms like polar coordinates.

Angle Calculation

Calculating the angle \(\theta\) in polar coordinates is essential for converting from rectangular coordinates. This involves using the arctangent function, as \(\theta\) represents the angle between the positive x-axis and the line connecting the origin to the point. To find this angle, you start with the formula:

• \(\tan\theta = \frac{y}{x}\),

where \(x\) and \(y\) are the rectangular coordinates.
For the point \((3, -\sqrt{3})\), we calculate:

• \(\tan\theta = \frac{-\sqrt{3}}{3} = -\frac{1}{\sqrt{3}}\).

From trigonometry, specifically in the unit circle context, this gives \(\theta = -\frac{\pi}{6}\).
However, angles are typically expressed within a specific range, such as from 0 to \(2\pi\). In this case, the angle \(-\frac{\pi}{6}\) must be adjusted to \(\frac{11\pi}{6}\), ensuring it lies within the desired range.

Radius Calculation

The radius \(r\) in polar coordinates symbolizes the distance from the origin to the point in question. Unlike rectangular coordinates, where a point's position is shown as a combination of x-y movements, polar coordinates use this radial distance and an angle. To compute \(r\), utilize the formula:

• \(r = \sqrt{x^2 + y^2}\).

This derivation comes straight from the Pythagorean theorem applied to the triangle formed by the point's projection on the axes.
Applying this to the coordinates \((3, -\sqrt{3})\):

• first, square the x-value: \(3^2 = 9\),
• square the absolute y-value: \((-\sqrt{3})^2 = 3\),
• then add these squares: \(9 + 3 = 12\),
• and take the square root: \(\sqrt{12} = 2\sqrt{3} \approx 3.464\).

Hence, the radius \(r\) for the point is approximately 3.464.

Coordinate Conversion

Coordinate conversion is the process of transforming a point's representation from one coordinate system to another. This is crucial in various fields of science and engineering where different types of coordinates provide advantages under specific circumstances.
To convert from rectangular to polar coordinates:

• Calculate the radius \(r\) as \(\sqrt{x^2 + y^2}\).
• Determine the angle \(\theta\) using \(\tan\theta = \frac{y}{x}\) and make necessary range adjustments.

For example, the rectangular coordinates \((3, -\sqrt{3})\) convert to polar coordinates by first finding \(r = 3.464\) and \(\theta = \frac{11\pi}{6}\), offering a different but equivalent way to describe the point's position.
This conversion is particularly useful for applications involving rotational dynamics or where calculations are more naturally expressed in polar form, such as with waves and rotational motion.