Problem 3 - Calculus Early Transcendentals

Question

\(3-4\) Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. $$ \begin{array}{lll}{\text { (a) }(1, \pi)} & {\text { (b) }(2,-2 \pi / 3)} & {\text { (c) }(-2,3 \pi / 4)}\end{array} $$

Step-by-Step Solution

Answer

(a) (-1, 0), (b) (-1, -√3), (c) (√2, -√2)

Step 1: Understanding Polar Coordinates

Polar coordinates are expressed in the form \(r, \theta\), where \( r\) is the radial distance from the origin and \(\theta\) is the angle measured in radians from the positive x-axis. Our task involves plotting these points and finding their Cartesian coordinates.

Step 2: Cartesian Coordinates from Polar (a)

For point (a) \(1, \pi\), the radial distance is 1 and the angle is \ \pi \ radians. Use the formulas \( x = r \cos \theta\) and \( y = r \sin \theta\) to convert to Cartesian. Here, \(x = 1 \times \cos\pi = -1\) and \(y = 1 \times \sin\pi = 0\). So, the Cartesian coordinates are \(-1, 0\).

Step 3: Cartesian Coordinates from Polar (b)

For point (b) \(2, -\frac{2\pi}{3}\), the radial distance is 2 and the angle is \ -\frac{2\pi}{3} \ radians. Using the conversion formulas: \(x = 2 \times \cos(-\frac{2\pi}{3}) = -1\) and \(y = 2 \times \sin(-\frac{2\pi}{3}) = -\sqrt{3}\). Thus, the Cartesian coordinates are \(-1, -\sqrt{3}\).

Step 4: Cartesian Coordinates from Polar (c)

For point (c) \(-2, \frac{3\pi}{4}\), the radial distance is -2, which means the point lies in the opposite direction. The angle is \ \frac{3\pi}{4} \ radians. Converting, \(x = -2 \times \cos(\frac{3\pi}{4}) = \sqrt{2}\) and \(y = -2 \times \sin(\frac{3\pi}{4}) = -\sqrt{2}\). Therefore, the Cartesian coordinates are \(\sqrt{2}, -\sqrt{2}\).

Key Concepts

Cartesian coordinates

The concept of Cartesian coordinates is foundational in much of mathematics and science. These coordinates define a point in space using a pair of numbers \(x, y\). Here, \(x\) represents the horizontal distance from the origin, and \(y\) corresponds to the vertical distance. The origin is at \(0, 0\), serving as the reference point for all other coordinates in the plane.
Cartesian coordinates are particularly useful because they provide a straightforward way to describe any location on a plane. Given any point, just two numbers can precisely communicate its position, allowing for ease in calculation and measurement.
This system is essential for graphing functions and equations, turning abstract algebraic expressions into geometric visuals that are easier to interpret and analyze. To master Cartesian coordinates, it helps to get comfortable moving back and forth between physical points on a graph and their corresponding \(x, y\) values. This skill creates a firm foundation for further study in topics like calculus and physics.

coordinate conversion

Coordinate conversion is the process of translating one form of coordinates into another. In this context, we’re discussing conversion between polar and Cartesian coordinates. Since both systems provide different ways to define points on a plane, being able to switch between them is incredibly useful.
Polar coordinates are given as \( r, \theta \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis. To convert polar coordinates into Cartesian, we apply the formulas:

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

These equations allow us to find how far to travel horizontally and vertically to reach the point from the origin. Conversely, to switch from Cartesian back to polar coordinates, use:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \arctan(\frac{y}{x}) \)

Mastering these conversions is crucial for solving problems that naturally fit into one system over the other due to their symmetry or context, such as circular motion problems in physics.

trigonometry

Trigonometry is a powerful branch of mathematics, dealing primarily with triangles, especially right triangles. Central to trigonometry are the functions sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), which relate angles to the ratios of different sides of a triangle.
These functions play a significant role in the conversion from polar to Cartesian coordinates. Specifically, knowing the angle \( \theta \) and applying \(\cos\) and \(\sin\) allows us to determine the horizontal and vertical distances (\(x\) and \(y\), respectively) of a point from the origin.
Triangular relationships are often depicted in the unit circle, where a circle with radius 1 centered at the origin provides a clear path to visualize these trigonometric functions. Here, the \(x\) value of a point on the circle corresponds to \(\cos\theta\), and the \(y\) value to \(\sin\theta\).
Understanding these concepts helps not only in coordinate conversion but also broadens the scope in various applications across science and engineering. Trigonometry simplifies the study of periodic functions and waves, making it indispensable within fields ranging from acoustics to electromagnetism.