Problem 32 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r^{2}=4 r \sin \theta $$

Step-by-Step Solution

Answer

The Cartesian equation is \(x^2 + (y - 2)^2 = 4\), a circle centered at \((0, 2)\) with radius 2.

Step 1: Identify the Given Polar Equation

We are given the polar equation \( r^2 = 4r \sin \theta \). Our goal is to convert this equation into a Cartesian form.

Step 2: Substitute Polar to Cartesian Conversion Formulas

Recall that in polar coordinates, the relationships with Cartesian coordinates \( (x, y) \) are given by \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Substitute \( r \sin \theta = y \) into the equation to get \( r^2 = 4y \).

Step 3: Simplify the Equation

We already have \( r^2 = x^2 + y^2 \). Hence, substitute this into the equation \( r^2 = 4y \), resulting in \( x^2 + y^2 = 4y \).

Step 4: Reorganize into Standard Form

Rearrange \( x^2 + y^2 = 4y \) by completing the square for the \( y \)-terms: \( x^2 + (y^2 - 4y) = 0 \). Complete the square: \( (y - 2)^2 - 4 \). Equation becomes \( x^2 + (y - 2)^2 = 4 \).

Step 5: Identify the Graph Type

The equation \( x^2 + (y - 2)^2 = 4 \) is in the standard form of a circle \( (x - h)^2 + (y - k)^2 = r^2 \), with center \((0, 2)\) and radius \(2\).

Key Concepts

Polar Equations

In mathematics, polar equations are equations that describe a curve using the polar coordinate system. Polar coordinates are defined with a radius, denoted as \( r \), and an angle, \( \theta \). This system is particularly useful for situations where relationships are better expressed in terms of angles and distances from a central point rather than \( x \) and \( y \) coordinates.

Polar equations usually express \( r \) as a function of \( \theta \), or vice versa. However, they can also include equations where \( r \) is equated to a trigonometric function, such as \( r = k \sin \theta \). In this context, polar equations often describe curves like spirals and roses.

• Coordinates are represented as \( (r, \theta) \).
• Relationship to Cartesian: \( x = r \cos \theta \), \( y = r \sin \theta \), \( r^2 = x^2 + y^2 \).

When working with polar equations, a common task is to convert them into the widely used cartesian form, which can help in identifying familiar graph shapes or simplifying calculations.

Cartesian Equations

Cartesian equations are a way to express geometrical shapes using the Cartesian coordinate system (\( x, y \) plane). These equations are formed with x and y variables representing horizontal and vertical positions, respectively. Cartesian equations are extensively used because of their simplicity in geometry and algebra.

Converting from polar to cartesian involves substituting polar equations with their cartesian equivalents. This includes using:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

For instance, transforming the equation \( r^2 = 4r \sin \theta \) into Cartesian form starts by substituting \( r \sin \theta = y \), leading to \( r^2 = 4y \). Using the identity \( r^2 = x^2 + y^2 \), we then find \( x^2 + y^2 = 4y \). From this, we can manipulate the terms to find a recognizable equation form.

Circle Equations

Circle equations represent circles on the coordinate plane. Typically, a circle with a center at \( (h, k) \) and a radius \( r \) can be expressed in the Cartesian standard form as \( (x - h)^2 + (y - k)^2 = r^2 \). This expression allows identifying the center and radius of the circle directly from the equation itself.

In the original exercise, we converted the polar equation \( r^2 = 4r \sin \theta \) to the Cartesian equation \( x^2 + (y - 2)^2 = 4 \). By completing the square, we've put it in the form of a circle equation:

• Center: \( (0, 2) \)
• Radius: \( 2 \)

Understanding how to work with these equations is crucial for describing and graphing geometrical figures. Converting equations into this form helps in identifying the shape and size of the circle quickly and accurately.