Problem 28 - Precalculus Mathematics for Calculus

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$(6,2 \pi / 3)$$

Step-by-Step Solution

Answer

The rectangular coordinates are (-3, 3√3).

Step 1: Understanding the Polar Coordinates

Given the polar coordinates (6, \frac{2\pi}{3}), where 6 is the radius \( r \) and \( \frac{2\pi}{3} \) is the angle \( \theta \). The goal is to convert these into rectangular coordinates \((x, y)\).

Step 2: Applying Conversion Formulas

To find the rectangular coordinates, we use the formulas: 1. \( x = r \cdot \cos(\theta) \)2. \( y = r \cdot \sin(\theta) \).Here, \( r = 6 \) and \( \theta = \frac{2\pi}{3} \).

Step 3: Calculating the x-coordinate

Compute \( x \):\[ x = 6 \cdot \cos\left(\frac{2\pi}{3}\right) \].Since \( \frac{2\pi}{3} \) is in the second quadrant, where cosine is negative, \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \). Thus,\[ x = 6 \cdot (-\frac{1}{2}) = -3 \].

Step 4: Calculating the y-coordinate

Compute \( y \):\[ y = 6 \cdot \sin\left(\frac{2\pi}{3}\right) \].The sine of \( \frac{2\pi}{3} \) is equal to \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), because sine is positive in the second quadrant. Therefore,\[ y = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \].

Step 5: Writing the Rectangular Coordinates

The rectangular coordinates are \((x, y) = (-3, 3\sqrt{3})\).

Key Concepts

Polar Coordinates

Imagine a point on a flat surface described using a pole (origin) and an angle. This system is called polar coordinates. It differs from the rectangular coordinate system, which uses horizontal and vertical measurements. In polar coordinates, a point is defined by:

• A distance from the pole (radius, \( r \))
• An angle (\( \theta \)) measured counterclockwise from a fixed direction (usually the positive x-axis)

This way of describing a point is particularly useful in situations where direction and distance from a central point are more intuitive than at a right angle layout. For example, radar and sonar systems often use polar coordinates.

Understanding polar coordinates helps in analyzing circles and spirals, and provides an easier approach to problems involving circular motion or other radially-symmetric phenomena. To fully utilize polar coordinates, being familiar with the angles in radians, such as \(\frac{2\pi}{3} \), is important. This helps not only in understanding the angle itself but also in relation to the quadrants of the coordinate system.

Coordinate Conversion

Converting from polar to rectangular coordinates involves using specific trigonometric relationships. The basic formulas to convert a polar coordinate \((r, \theta)\) to a rectangular form \((x, y)\) are:

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

These formulas rely on the trigonometric functions cosine and sine, which project the radius \( r \) onto the x and y axes respectively.

Begin conversion by identifying the angle's quadrant—this is crucial because it determines the sign of \(\cos(\theta)\) and \(\sin(\theta)\). In this exercise, since \(\theta = \frac{2\pi}{3}\) is in the second quadrant, \(\cos\) is negative and \(\sin\) is positive.

Accurately calculating using these properties ensures that the converted coordinates, such as \(-3\) and \(3\sqrt{3}\) in this example, accurately reflect the original point’s position.

Trigonometry

Trigonometry is the study of relationships between angles and sides in triangles. It provides the tools needed for coordinate conversion, specifically in its functions sine and cosine. These functions relate an angle in a right triangle to the lengths of its sides.

In the context of converting polar to rectangular coordinates, understanding these trigonometric functions is essential. When we compute \( x = 6 \cdot \cos\left(\frac{2\pi}{3}\right) \), we rely on the cosine function to project the length onto the x-axis. Similarly, \( y = 6 \cdot \sin\left(\frac{2\pi}{3}\right) \) uses sine to find the y-coordinate contribution.
The values \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \) and \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \) derive from understanding the unit circle, a circle with a radius of 1 centered at the origin. In this circle, angles are measured in radians, and the x-coordinate of any angle represents \( \cos(\theta) \), while the y-coordinate corresponds to \( \sin(\theta) \).

Mastery of these concepts is beneficial not just for coordinate conversion but also for various applications in physics, engineering, and any field involving wave patterns or circular motion.