Problem 8 - Thomas Calculus

Question

Find the polar coordinates, \(-\pi \leq \theta<\pi\) and \(r \geq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(-2,-2)} & {\text { b. }(0,3)} \\ {\text { c. }(-\sqrt{3}, 1)} & {\text { d. }(5,-12)}\end{array}$$

Step-by-Step Solution

Answer

a: \\(2\sqrt{2}, -\frac{3\pi}{4}\\); b: \\(3, \frac{\pi}{2}\\); c: \\(2, \frac{5\pi}{6}\\); d: \\(13, -1.176\\) (radians).

Step 1: Understand the Relationship between Cartesian and Polar Coordinates

Polar coordinates \(r, \theta\) are related to Cartesian coordinates \(x, y\) through the following formulas: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). If the Cartesian coordinate \(x\) is zero, the angle can directly be considered to lie on the y-axis with adjustments for sign.

Step 2: Convert Point "a" (-2,-2) to Polar Coordinates

For point \(a\) with \(x = -2\) and \(y = -2\):1. Calculate \(r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}\).2. Calculate initial \(\theta = \tan^{-1}\left(\frac{-2}{-2}\right) = \tan^{-1}(1) = \frac{\pi}{4}\).3. Adjust \(\theta\) to the correct quadrant. Since both \(x\) and \(y\) are negative, \(\theta\) lies in the third quadrant: \(\theta = \frac{\pi}{4} - \pi = -\frac{3\pi}{4}\).4. Therefore, the polar coordinates are \((2\sqrt{2}, -\frac{3\pi}{4})\).

Step 3: Convert Point "b" (0, 3) to Polar Coordinates

For point \(b\) with \(x = 0\) and \(y = 3\):1. Calculate \(r = \sqrt{0^2 + 3^2} = 3\).2. Since \(x = 0\) and \(y>0\), the angle \(\theta\) is \(\frac{\pi}{2}\) (point is located on the positive y-axis).3. Therefore, the polar coordinates are \((3, \frac{\pi}{2})\).

Step 4: Convert Point "c" (-\sqrt{3}, 1) to Polar Coordinates

For point \(c\) with \(x = -\sqrt{3}\) and \(y = 1\):1. Calculate \(r = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2\).2. Calculate initial \(\theta = \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right) = \tan^{-1}(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}\).3. Adjust \(\theta\) to the correct quadrant. Since \(x < 0\) and \(y > 0\), \(\theta\) is in the second quadrant: \(\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\).4. Therefore, the polar coordinates are \((2, \frac{5\pi}{6})\).

Step 5: Convert Point "d" (5,-12) to Polar Coordinates

For point \(d\) with \(x = 5\) and \(y = -12\):1. Calculate \(r = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = 13\).2. Calculate initial \(\theta = \tan^{-1}\left(\frac{-12}{5}\right) = \tan^{-1}(-2.4)\).3. Since \(\tan^{-1}(-2.4)\) provides an angle to the fourth quadrant, \(\theta = -1.176 ext{ radians}\) (approximated), and it's within given limits (\(-\pi \leq \theta < \pi\)).4. Therefore, the polar coordinates are \((13, -1.176)\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a common way to describe a point in a two-dimensional plane. Each point is defined by a pair of numbers \( x \) and \( y \), which represent distances along horizontal and vertical axes, respectively. Think of Cartesian coordinates like your typical grid system on a map.

• The horizontal line, known as the x-axis, is used to determine the left-right position.
• The vertical line, called the y-axis, specifies the up-down location.
• The origin is the point where both axes intersect, represented by \( (0,0) \).

In mathematical and everyday contexts, Cartesian coordinates help locate points easily when working with linear movements or geometric figures within a plane.
Cartesian coordinates form the foundation for many areas of mathematics, including geometry and algebra, by providing a clear way to visualize relationships between points.

Coordinate Conversion

Converting between Cartesian and polar coordinates involves a few simple yet crucial steps. This process requires using trigonometric relationships and basic operations.

• To find the radial coordinate \( r \), use the formula \( r = \sqrt{x^2 + y^2} \). This is derived from the Pythagorean theorem, as it finds the distance from the origin to the point.
• The angular coordinate \( \theta \), which makes up the polar coordinates, can be calculated using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).
• Adjusting \( \theta \) is necessary to ensure it is correctly placed within the proper quadrant, especially when \( x \) or \( y \) is negative.

This conversion is essential for switching between the rectangular grid layout of Cartesian coordinates and the angle-direction oriented polar system.
Having a grasp on coordinate conversion allows for versatility in solving mathematical problems, making it easier to switch systems based on what is most practical for the problem at hand.

Trigonometry in Polar Coordinates

In polar coordinates, trigonometry plays a key role in determining the direction and angle of points. Unlike Cartesian coordinates that primarily use linear distances, polar coordinates leverage the angle from a central point, or origin.

• The angle \( \theta \) represents the counterclockwise rotation from the positive x-axis to the point.
• Trigonometric functions such as tangent, sine, and cosine help in defining this angle and converting it back to Cartesian coordinates if needed.
• The sine function, for example, relates to the y-component while cosine relates to the x-component when recalculating back to Cartesian.

Understanding the trigonometry involved in polar coordinates allows for accurate descriptions of circles, rotations, and spirals.
Polar coordinates offer a more intuitive and natural way to describe situations where direction and distance from a central point are more pertinent than grid-like positioning.