Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x^{2}+y^{2}=4\)

Step-by-Step Solution

Answer

The polar equation is \(r = 2\).

Step 1: Understand the Cartesian Equation

The given Cartesian equation is \(x^2 + y^2 = 4\). This is the equation of a circle centered at the origin (0,0) with a radius of 2, because the equation \(x^2 + y^2 = r^2\) describes a circle where \(r\) is the radius.

Step 2: Sketch the Cartesian graph

Draw a circle centered at the origin with a radius of 2 on the Cartesian coordinate plane. This circle will touch the points (2,0), (-2,0), (0,2), and (0,-2).

Step 3: Convert the Equation to Polar Coordinates

In polar coordinates, relationships are given by \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the Cartesian equation: \(r^2 \cos^2 \theta + r^2 \sin^2 \theta = 4\).

Step 4: Simplify the Polar Equation

From the identity \(\cos^2 \theta + \sin^2 \theta = 1\), we can rewrite the equation as \(r^2(\cos^2 \theta + \sin^2 \theta) = r^2 \cdot 1 = r^2\). Thus, the equation becomes \(r^2 = 4\).

Step 5: Solve for the Polar Radius

Taking the square root of both sides, we find \(r = 2\). This means the polar equation describing the same circle is \(r = 2\), where the radius is constant for all angles \(\theta\).

Key Concepts

Cartesian Equation

A Cartesian equation is one that defines a relationship between the x and y coordinates on a Cartesian coordinate system, which is structured like a grid. Each point on the grid can be described by a pair of numbers \( (x, y) \). This is often used to describe lines, curves, and other shapes.
One of the simplest forms is the equation of a line, \((y = mx + c)\), but when it comes to curves, circles are very common.
The general Cartesian equation of a circle is \( x^2 + y^2 = r^2 \).
Here, \( r \) is the radius of the circle, and the origin \( (0, 0) \) is the center.

• To identify a circle in a Cartesian equation, look for \( x^2 \) and \( y^2 \) terms with the same positive coefficients.
• The sum of these squared terms equals a constant, which will be the square of the radius.

Circle Equation

The equation of a circle in Cartesian coordinates is a specific type of quadratic equation. It is represented as \( x^2 + y^2 = r^2 \).
This indicates a circle centered at the origin with a radius \( r \). For instance, the equation \( x^2 + y^2 = 4 \) describes a circle with a radius of 2.
This is because \( 4 = 2^2 \). An essential step in working with circle equations is understanding how to find the radius and center.

• If the circle is shifted, the general form becomes \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center.
• The radius \( r \) dictates how far from the center the circle extends.

Such equations allow us to determine the geometric shape and size of a circle within a coordinate plane.

Coordinate Conversion

Converting between coordinate systems—specifically from Cartesian to polar coordinates—allows for a more flexible representation of shapes and equations.
This is particularly useful for circles and curves. In polar coordinates, any point \( (x, y) \) in the Cartesian system can be represented using \( (r, \theta) \).

• Here, \( r \) is the distance from the origin to the point, and \( \theta \) is the angle from the positive x-axis to the line connecting the origin to the point.
• The relationships \( x = r \cos \theta \) and \( y = r \sin \theta \) help convert Cartesian equations to their polar counterparts.

For example, starting with the circle equation \( x^2 + y^2 = 4 \), we substitute to get \( r^2 \cos^2 \theta + r^2 \sin^2 \theta = 4 \). Knowing that \( \cos^2 \theta + \sin^2 \theta = 1 \), this simplifies neatly into \( r^2 = 4 \). By taking the square root, we get \( r = 2 \), which is our polar equation for the circle.

Graph Sketching

Graph sketching is about drawing the geometric representation of equations.
This process helps to visualize mathematical solutions and understand their implications. When sketching a graph, it's crucial to recognize the shape depicted by the equation.
A circle, for example, is identified by its equation form.

• First, identify key characteristics like the center and radius of the circle \(x^2 + y^2 = 4\).
• This circle centers at the origin and has a radius of 2.
• Mark this on the graph by plotting points (2,0), (-2,0), (0,2), and (0,-2) on the Cartesian plane.

Join these points smoothly to form a circle.
For polar coordinates, graphing can further reveal circular symmetry around the origin.
A polar equation like \( r = 2 \) suggests that at any angle \( \theta \), the circle's radius remains constant.
Understanding such visual techniques is crucial to mastering graph representation of mathematical concepts.