Problem 42 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

The polar form of the equation is \( r = 3 \).

Step 1: Understand the Cartesian equation

The given equation is in the Cartesian form: \( x^2 + y^2 = 9 \). It represents a circle with a radius of 3 centered at the origin.

Step 2: Recall polar coordinates

In polar coordinates, the relationship between Cartesian and polar coordinates is given by: \( x = r \cos \theta \) and \( y = r \sin \theta \). The distance from the origin is \( r = \sqrt{x^2 + y^2} \).

Step 3: Substitute in polar coordinates

Substitute the polar coordinate relationships into the Cartesian equation. Since \( x^2 + y^2 = r^2 \), we have \( r^2 = 9 \).

Step 4: Solve for r

Solve the equation \( r^2 = 9 \) for \( r \). Taking the square root gives \( r = \pm 3 \). However, since \( r \) represents a radius or distance in polar coordinates, it must be non-negative. Thus, \( r = 3 \).

Step 5: Conclude the polar form

The polar form of the equation representing the circle is \( r = 3 \). This equation remains true for any angle \( \theta \), indicating a constant radius of 3.

Key Concepts

Conversion of Equations

Converting equations from one form to another is a fundamental skill in mathematics. This process allows us to view and analyze the same problem from different perspectives, which can often simplify complex problems or provide new insights.

To convert a Cartesian equation, such as \( x^2 + y^2 = 9 \), into polar form, we rely on the polar coordinate system's relationship with Cartesian coordinates. In essence, this conversion involves substituting polar coordinate expressions for \( x \) and \( y \) into the original equation.

This approach utilizes the identities \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( r \) is the radial distance from the origin, and \( \theta \) is the angle from the positive x-axis. By substituting these expressions into the Cartesian equation, we effectively transform it into polar coordinates.

Cartesian Form

The Cartesian form of an equation is expressed in terms of \( x \) and \( y \) coordinates. This form is familiar and widely used, especially in geometric contexts. The given equation \( x^2 + y^2 = 9 \) is an example of a Cartesian form, which represents a circle with a radius of 3 centered at the origin \((0,0)\).

The Cartesian form is particularly useful for straightforward visualization and graphing on a standard coordinate plane, known as the Cartesian plane. Here, the axes represent the x and y values of points.

Understanding the Cartesian form as a foundation is essential before transitioning to other coordinate systems, such as polar coordinates, where the concept of location is approached differently.

Polar Form

The polar form of an equation changes how we understand the spatial relationship between points. In this form, each point is described by a radius and an angle. The equation \( r = 3 \) is the polar form of the circle given in Cartesian form as \( x^2 + y^2 = 9 \).

This form is particularly effective for circular shapes and periodic functions, facilitating a natural description of phenomena involving rotation or wave patterns. By using polar coordinates, some problems become easier to solve or visualize, as relationships are expressed in terms of direction and distance.

In polar form, the radius \( r \) remains constant for circles. Thus, \( r = 3 \) represents a circle of radius 3, with every point on the circle being 3 units from the origin regardless of the angle \( \theta \).

Trigonometric Identities

Trigonometric identities play a crucial role in the conversion of equations from Cartesian to polar form. They are the bridge that allows us to switch between the two systems using known relationships between angles and sides of triangles.

The identities \( x = r \cos \theta \) and \( y = r \sin \theta \) are derived from the definitions of cosine and sine in a right triangle. These equations express how we can decompose the radial distance, \( r \), into its horizontal and vertical components in terms of \( x \) and \( y \) respectively.

Utilizing these identities allows us to substitute back and forth between systems, making problems more manageable. Whether working with wave functions, angular motion, or simply graphing geometric shapes, understanding and applying these identities is indispensable.