Problem 65 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(x^2+y^2=9\)

Step-by-Step Solution

Answer

The polar form of the given equation is \(r^2 = 9\).

Step 1: Identify the given rectangular equation

The given rectangular equation to convert to polar form is \(x^2+y^2=9\).

Step 2: Write down the relationships between rectangular and polar coordinates

In terms of polar coordinates, \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Hence \(x^2 = r^2\cos^2(\theta)\) and \(y^2 = r^2\sin^2(\theta)\).

Step 3: Substitute the rectangular coordinates with the polar coordinates in the equation

Substitute the values of \(x^2\) and \(y^2\) from step 2 into the equation. Hence we get \(r^2\cos^2(\theta) + r^2\sin^2(\theta) = 9\).

Step 4: Simplify the equation

The sum of the squares of sin and cos of the same angle is always equal to 1. Therefore, \(r^2(\cos^2(\theta) + \sin^2(\theta))\) simplifies to \(r^2 * 1\), which is \(r^2\). Set this equal to 9 and we obtain the equation in polar form, which is \(r^2 = 9\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent points in a plane using cartesian coordinates, typically denoted as \((x, y)\). These coordinates describe a location by providing horizontal \((x)\) and vertical \((y)\) distances from a fixed reference, or origin, point. They are particularly intuitive because they reflect our common navigation in space, like moving left, right, up, and down.
This system is highly useful for plotting graphs, solving geometrical problems, and expressing mathematical functions.
In our exercise, the equation \(x^2 + y^2 = 9\) is given in rectangular form. This type of equation is often encountered as part of the study of circles, where the equation represents all points distance of 3 units from the origin in the plane.

Conversion

Converting from one form of coordinates to another is crucial in mathematics for simplifying problems or gaining different insights into problems. Here, we are converting from rectangular coordinates to polar coordinates.
The key relationships to remember in this conversion are:

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

These relationships allow you to express the \(x\) and \(y\) coordinates in terms of \(r\), the distance from the origin, and \(\theta\), the angle made with the positive x-axis. Just substitute these into your original rectangular equation to get a polar equation.
In the equation given, \(x^2 + y^2 = 9\), substitute for \(x\) and \(y\) with their polar counterparts, summed together utilizing the identity \(cos^2(\theta) + sin^2(\theta) = 1\) to further simplify.

Mathematical Equations

Mathematical equations are expressions that assert the equality of two expressions. They play a pivotal role in understanding various concepts of mathematics as they allow for precise computation and analysis of quantities and figures.
In this specific problem, we start with a standard form of a circle equation in rectangular coordinates, \(x^2 + y^2 = 9\). By converting this equation into the polar form equation \(r^2 = 9\), we gain insight into the representation of this circle in a different coordinate system.
Understanding these equations in polar form is particularly useful when dealing with problems involving rotations or when symmetry makes polar coordinates a more natural choice. In polar form, the equation \(r^2 = 9\) indicates that every point \(r\) is 3 units away from the origin, confirming the depiction of a circle of radius 3.