Problem 136 - Calculus Volume 3

Question

The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2?]. Round to three decimal places. $$ (2,2) $$

Step-by-Step Solution

Answer

The polar coordinates are (2.828, 0.785) and (2.828, 7.068).

Step 1: Understand the Relationship Between Cartesian and Polar Coordinates

Cartesian coordinates (2,2) need to be converted to polar coordinates where each point is expressed in the form \((r, \theta)\). Here, \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

Step 2: Calculate the Distance r

The distance \(r\) is calculated using the formula \[ r = \sqrt{x^2 + y^2} \]Substitute \(x = 2\) and \(y = 2\):\[ r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \approx 2.828 \]

Step 3: Calculate the Angle θ for the First Set of Coordinates

The angle \(\theta\) is calculated using \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]Substitute \(x = 2\) and \(y = 2\):\[ \theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4} \approx 0.785 \] radians.

Step 4: Determine the Second Set of Polar Coordinates

In polar coordinates, adding \(2\pi\) radians to \(\theta\) results in an equivalent angle due to periodicity. Thus, another set of polar coordinates is\[ \theta + 2\pi = \frac{\pi}{4} + 2\pi \approx 0.785 + 6.283 = 7.068 \] radians.

Step 5: Formulate the Two Sets of Polar Coordinates

Hence, two sets of polar coordinates for the given point are:- First set: \((2.828, 0.785)\)- Second set: \((2.828, 7.068)\)

Key Concepts

Cartesian Coordinates

Cartesian coordinates represent a specific point in the plane based on two values: the horizontal component (x) and the vertical component (y). These coordinates are often referred to as rectangular coordinates and are widely used for graphing on a two-dimensional plane. In the exercise, we are given Cartesian coordinates (2,2), which can be visualized as a point that is two units along the x-axis and two units vertically up the y-axis. This forms a square-like structure when plotting points that share similarities, making calculations straightforward. Since the Cartesian system uses perpendicular lines (axes) to calculate distances and directions, it provides a clear and direct way to locate points in a plane. However, when you need to express points in terms of magnitude and direction, converting them into polar coordinates becomes necessary.

Conversion Between Coordinate Systems

Converting between Cartesian and polar coordinates involves using specific mathematical formulas. In polar coordinates, a point is represented as \((r, \theta)\) where \(r\) is the radial distance from the origin, and \(\theta\) is the angular displacement from the positive x-axis. The conversion process helps in situations where angles and distances from the origin need better illustration.

• The radial distance \(r\) is calculated using the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). In our case, substituting \(x = 2\) and \(y = 2\) gives \(r \approx 2.828\).

• To find \(\theta\), you can use the inverse tangent function: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). Thus, \(\theta = \tan^{-1}(1) = \frac{\pi}{4} \approx 0.785\) radians.

This process illustrates how distances and angles relate to point locations in different coordinate systems.

Trigonometry

Trigonometry is fundamental in determining the relationship between angles and distances. It plays a critical role while transforming between Cartesian and polar coordinates. Using trigonometric relationships, you can determine \(\theta\) by applying the arctangent (\(\tan^{-1}\)) function, which helps find the angle given a ratio of the sides of a right triangle.A clear understanding of trigonometric functions allows you to visualize how angles—and thus directions—correspond to spatial layouts. Though initially complex, these functions simplify the work by providing insights into how geometric figures behave, particularly when requiring conversions across coordinate systems, like moving from Cartesian to polar or vice versa.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebraic techniques and geometric concepts to understand spatial relationships. The exercise demonstrates this by analyzing a given point (2,2) and converting it into its polar equivalent.Understanding the geometry behind these coordinate systems reveals how shapes and points behave on graphs. Polar coordinates inherently provide a different perspective by focusing on distances and angles, offering unique ways to approach geometry problems. Unlike the traditional rectangular grid visible in Cartesian systems, polar coordinates create circular plots emphasizing radial symmetry. This shift in perspective can be particularly advantageous when solving problems involving rotations and periodic functions, as evident when deriving the second set of polar coordinates by adding \(2\pi\) to the initial \(\theta\). Through these methods, students can handle a multitude of mathematical problems efficiently by assessing the most suitable coordinate system for their specific needs.