Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(3, \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The Cartesian coordinates of the point \((3, \frac{5\pi}{6})\) are \(-(3\sqrt{3}/2, 3/2)\). The three additional polar coordinates representing the same point are \((3, -\frac{1\pi}{6})\), \((3, \frac{7\pi}{6})\), and \((3, \frac{3\pi}{2})\) respectively.

Step 1: Conversion to Cartesian Coordinates

First, convert the original polar point to Cartesian coordinates using the definitions \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). So we substitute \(r = 3\) and \(\theta = \frac{5\pi}{6}\), and we get \(x = 3 \cos(\frac{5\pi}{6})\) and \(y = 3 \sin(\frac{5\pi}{6})\). After computation, we obtain the Cartesian coordinates.

Step 2: Beginning to Find Polar Representations

For a point in polar coordinates \((r,\theta)\), the point \((r, \theta + 2n \pi)\) is the same for any integer n. Therefore, adding multiples of \(2\pi\) to the angle will provide additional polar coordinates.

Step 3: Find three additional polar coordinates

Now, find three additional polar representations by letting \(n = -1, 1, 2\). This leads us to \(\theta = \frac{5\pi}{6} + 2(\pi)(-1) = -\frac{1\pi}{6}\), \(\theta = \frac{5\pi}{6} + 2(\pi)(1) = \frac{7\pi}{6}\), and \(\theta = \frac{5\pi}{6} + 2(\pi)(2) = \frac{3\pi}{2}\). So the three additional polar coordinates representing the same point are \((3, -\frac{1\pi}{6})\), \((3, \frac{7\pi}{6})\) and \((3, \frac{3\pi}{2})\) respectively.

Key Concepts

Cartesian coordinates

In a two-dimensional space, the Cartesian coordinate system is a way of specifying the position of points using two numbers. These numbers define the horizontal and vertical distances of a point from a fixed reference point, known as the origin. Imagine a grid where each point is defined by a pair \(x, y\).
From our original exercise, we are starting with polar coordinates, which will need to be converted to Cartesian coordinates as a part of the solution. When given polar coordinates \(r, \theta\), conversion involves the use of trigonometric functions:

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

These formulas help translate the circular motion concept in polar coordinates into a more traditional linear grid seen in Cartesian coordinates. By substituting the necessary values, you can see how a point moves from one system of representation to the other. This conversion process is crucial for fields like physics and engineering, where different representations may simplify problem solving.

polar representation

Polar coordinates provide a different method for defining a point's location. Instead of defining a point by its horizontal and vertical distance from the origin, polar coordinates use a distance from the origin and an angle from a reference direction.
Polar coordinates are noted as (r, \theta), where:

• \(r\) is the radius or the distance of the point from the origin.
• \(\theta\) is the angle measured from the positive x-axis (in a counter-clockwise direction).

In the given exercise, a point is described by the polar coordinates \(3, \frac{5\pi}{6}\). To find alternate forms of this point, we utilize the property that a rotation by full circles (2\pi radians) results in the same position. Hence, altering \(\theta\) by multiples of \(2\pi\), while keeping \(r\) fixed, reveals several representations of the same point—a useful property when solving complex geometry problems.

angle conversion

Converting angles between different systems or different values is a useful aspect of mathematics and physics. In the context of polar coordinates, understanding how to manipulate angles is essential.
For the exercise, consider angle conversion by adding or subtracting multiples of \(2\pi\) from \(\theta\). This step aims to identify various coterminal angles that share the same position:

• Given \(\theta = \frac{5\pi}{6}\), adding \(2\pi\times n\) (where \(n\) is any integer) results in equivalent polar coordinates.

For our solution, using \(n = -1, 1,\) and \(2\), different angle representations of the same point were calculated.

• \(\theta = \frac{5\pi}{6} - 2\pi = -\frac{\pi}{6}\)
• \(\theta = \frac{5\pi}{6} + 2\pi = \frac{7\pi}{6}\)
• \(\theta = \frac{5\pi}{6} + 4\pi = \frac{3\pi}{2}\)

These calculations present a strategic advantage in fields such as robotics and navigation, where precise angle adjustments can ensure accurate positioning and direction.