Problem 61 - Precalculus with Limits

Question

In Exercises 55-64, use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. \(\left(\frac{5}{2}, \frac{4}{3}\right)\)

Step-by-Step Solution

Answer

The polar coordinates for the rectangular coordinates \(\left(\frac{5}{2}, \frac{4}{3}\right)\) are (r, θ) where r and θ are calculated values from the step 3.

Step 1: Identify the rectangular coordinates

The given rectangular coordinates are \(\left(\frac{5}{2}, \frac{4}{3}\right)\)

Step 2: Convert rectangular to polar coordinates

The conversion from rectangular to polar coordinates involves finding the radius (r), which is the distance to the point from the origin, and the angle (θ), which is the counterclockwise angle from the positive x-axis. The formula for finding r is \( r = \sqrt{xˆ2 + yˆ2} \). For θ, take the inverse tangent of \( y / x \) if x > 0, and \( π + y / x \) if x < 0. Here, x = \(\frac{5}{2}\) and y = \(\frac{4}{3}\), so, \( r = \sqrt{(\frac{5}{2})ˆ2 + (\frac{4}{3})ˆ2 } \), and since x is positive, \( θ = tanˆ(-1)(\frac{%}{%}) \).

Step 3: Calculate the polar coordinates

Substitute the values of x and y into the equations for r and θ to get the polar coordinates. \( r = \sqrt{(\frac{5}{2})ˆ2 + (\frac{4}{3})ˆ2 } \), \( θ = tanˆ(-1)(\frac{4/3}{5/2}) \). Calculate the values. r and θ represent the polar coordinates

Key Concepts

Polar Coordinates

When we talk about polar coordinates, we're referring to a two-dimensional coordinate system that presents locations on a plane using a radius and an angle. Unlike the more common rectangular coordinate system, which uses horizontal and vertical distances (x and y) to describe a point's position, polar coordinates use a direct distance from a fixed point, known as the origin, combined with an angle measured from a reference direction, typically the positive x-axis.

Specifically, a point expressed in polar coordinates is written as \( (r, \theta) \), where \( r \) stands for the radius - the distance from the origin - and \( \theta \) is the angle in radians, which shows how far around from the reference direction you have to rotate to look directly at the point. This system is particularly useful in fields such as physics, engineering, and navigation, where circular motion and rotational symmetry play a prominent role.

Rectangular Coordinates

In contrast, rectangular coordinates are the type of coordinate system most people are familiar with from algebra and geometry classes. This system is also known as Cartesian coordinates, named after the mathematician René Descartes. In this setup, we use two perpendicular axes: often called the x-axis (horizontal) and the y-axis (vertical). Each point in the plane is determined by an x-coordinate (horizontal distance from the origin) and a y-coordinate (vertical distance from the origin), usually written as \( (x, y) \).

Rectangular coordinates are exceptionally useful for describing points, lines, and shapes on a flat surface and are the basis of most graphing that students learn in school. When dealing with functions, data analysis, or geometry, rectangular coordinates provide a simple and straightforward method to represent and manipulate information.

Radius and Angle Calculation

Converting from rectangular to polar coordinates means you must calculate the radius (\( r \) and the angle (\( \theta \) based on the given x and y coordinates.

The radius is found using the Pythagorean theorem, which in the context of coordinate geometry, states that the distance from the origin to a point \( (x, y) \) is \( r = \sqrt{x^2 + y^2} \). This provides the length of the straight line from the origin to our point.

The angle, \( \theta \) which is measured in radians, is found using the arctangent function. If the x-coordinate is positive, \( \theta \) is simply the arctangent of \( \frac{y}{x} \). However, if the x-coordinate is negative, we find \( \theta \) by adding \( \pi \) to the arctangent of \( \frac{y}{x} \) to account for the point being on the left side of the y-axis. This adjustment is necessary because the arctangent function only outputs angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) (or -90 to 90 degrees), and we must take into account the full circle of 360 degrees (or \( 2\pi \) radians).

Graphing Utility

Today, converting between these two coordinate systems is made easy and error-proof by using a graphing utility. These tools can rapidly handle the calculations and can provide visual representations, which are incredibly helpful for ensuring accurate conversions. A graphing utility or calculator often allows a user to input a set of rectangular coordinates and receive the corresponding polar coordinates without manual calculation. These utilities play a crucial role in math and science education, making it more accessible for students to understand complex concepts and visualize how different coordinate systems represent the same geometric figures or data points.

Using graphing utilities also helps prevent errors in calculation. When dealing with fractional coordinates, like \(\left(\frac{5}{2}, \frac{4}{3}\right)\), a graphing utility simplifies the process and provides the radius and angle in decimal or exact form, depending on the settings. This accessibility to various tools ensures that students can focus on understanding the concepts rather than getting bogged down by challenging arithmetic.