Problem 43 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (-4,-4 \sqrt{3}) $$

Step-by-Step Solution

Answer

Polar coordinates: \((8, \frac{4\pi}{3})\)

Step 1: Understand the Rectangular Coordinates

The given rectangular coordinates are \((-4, -4\sqrt{3})\). In rectangular coordinates, the first value \(-4\) represents the x-coordinate and \(-4\sqrt{3}\) represents the y-coordinate.

Step 2: Calculate the Radius \(r\)

Using the formula for the radius in polar coordinates, \( r = \sqrt{x^2 + y^2} \), substitute the x and y values:\[ r = \sqrt{(-4)^2 + (-4\sqrt{3})^2} \]\[ r = \sqrt{16 + 48} \]\[ r = \sqrt{64} \]\[ r = 8 \]. The radius \( r \) is 8.

Step 3: Calculate the Angle \(\theta\)

Use the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) to find the angle.\[ \theta = \tan^{-1}\left(\frac{-4\sqrt{3}}{-4}\right) \]\[ \theta = \tan^{-1}(\sqrt{3}) \]The exact angle for \(\tan^{-1}(\sqrt{3})\) is \(\frac{\pi}{3}\), but since the point is in the third quadrant, adjust it by adding \(\pi\):\[ \theta = \pi + \frac{\pi}{3} \]\[ \theta = \frac{4\pi}{3} \]. The angle \(\theta\) is \(\frac{4\pi}{3}\).

Step 4: Write the Polar Coordinates

Combine the calculated \(r\) and \(\theta\) to write the polar coordinates: \((r, \theta) = (8, \frac{4\pi}{3})\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to describe the position of a point in a plane using two numerical values. These values are typically labeled as \(x\) and \(y\), representing the horizontal and vertical positions, respectively. In the original exercise, the point is given as \((-4, -4\sqrt{3})\). This means the point lies 4 units to the left of the origin along the x-axis and \(4\sqrt{3}\) units downward along the y-axis.
Such coordinates are frequently used in geometry and calculus as they provide a straightforward way to describe locations in two-dimensional space. Using rectangular coordinates, we can visually locate points in relation to the origin, the point where both x and y are zero.
The concept can be extended into three dimensions with the introduction of a z-coordinate, but for the purpose of this exercise, we are focusing on a two-dimensional plane.

Radius Calculation

The radius in polar coordinates is the distance from the origin to the point in the plane. It is calculated using the Pythagorean theorem. For the given point \((-4, -4\sqrt{3})\), we use the formula:

• \( r = \sqrt{x^2 + y^2} \)

By substituting the x and y values:

• \( x = -4 \)
• \( y = -4\sqrt{3} \)
• \( r = \sqrt{(-4)^2 + (-4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \)

The radius, \( r \), is a crucial part of the conversion to polar coordinates, as it provides the direct distance from the origin. It is always a non-negative value, which signifies its magnitude. Understanding this concept is essential for grasping how points translate between different coordinate systems.

Angle Calculation

To find the angle \(\theta\) in polar coordinates, we calculate it based on the ratio of the y-coordinate to the x-coordinate, using the arctangent function, denoted as \( \tan^{-1} \). The formula to find \(\theta\) is:

• \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)

For our example, \( y = -4\sqrt{3} \) and \( x = -4 \):

• \( \theta = \tan^{-1}\left(\frac{-4\sqrt{3}}{-4}\right) = \tan^{-1}(\sqrt{3}) \)

This gives an angle of \( \frac{\pi}{3} \), but we must adjust because the point is located in the third quadrant of the Cartesian plane. In this quadrant, both x and y values are negative.
To find the correct angle for the third quadrant, we add \(\pi\) to \(\frac{\pi}{3}\):

• \( \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \)

This adjustment ensures the angle correctly reflects the point's orientation relative to the positive x-axis.

Trigonometry

Trigonometry plays a vital role in converting between rectangular and polar coordinates. It involves the study of angles and the relationships between angle measurements and distance. Specifically, we employ trigonometric functions such as tangent, sine, and cosine to make these calculations.
For our coordinate conversion, we heavily rely on the **tangent function**. With the arctangent, we determine how the point's y-coordinate compares to its x-coordinate — essentially its slope. Calculating the angle in this manner aids in a precise translation from rectangular to polar form.
Furthermore, the Pythagorean theorem, a foundational principle of trigonometry, helps in determining the radius, or hypotenuse, in our polar system, further demonstrating the interconnections between these mathematical concepts.

Coordinate Conversion

Coordinate conversion is the process of transforming a point from one system to another. In this exercise, we translate from rectangular coordinates \((-4, -4\sqrt{3})\) to polar coordinates \((r, \theta)\).

• Start by finding the radius **r** using \( r = \sqrt{x^2 + y^2} \), which measures the point's distance from the origin.
• Next, calculate the angle **\(\theta\)** with \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \), adjusting based on the quadrant to ensure accuracy.
• We derived \((r, \theta) = (8, \frac{4\pi}{3})\), indicating exactly where the point lies on the plane's circle.

This conversion method is essential as each coordinate system offers unique advantages: rectangular for straightforward plotting and operations, and polar for scenarios involving rotations and symmetrical patterns around a central point. Mastery of these conversions opens new perspectives in analyzing and understanding geometric relationships.