Problem 17 - Calculus Early Transcendentals

Question

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=3 \sin \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is \( x^2 + y^2 - 3y = 0 \), representing a circle.

Step 1: Understand the Polar Equation

The polar equation given is \( r = 3 \sin \theta \). This is a representation of a curve using polar coordinates, where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle with the positive x-axis.

Step 2: Recall Conversion Formulas

To convert a polar equation to a Cartesian equation, we use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, note that \( r^2 = x^2 + y^2 \).

Step 3: Express \( r \) in Terms of \( y \)

Since \( r = 3 \sin \theta \), we can multiply both sides by \( r \) to obtain: \( r^2 = 3r \sin \theta \). We can replace \( r \sin \theta \) with \( y \), giving \( r^2 = 3y \).

Step 4: Substitute \( r^2 \) with \( x^2 + y^2 \)

Using the identity \( r^2 = x^2 + y^2 \), substitute into the equation from Step 3 to obtain: \( x^2 + y^2 = 3y \).

Step 5: Rearrange and Simplify

Rearrange the equation \( x^2 + y^2 = 3y \) to \( x^2 + y^2 - 3y = 0 \). This is the Cartesian equation of the curve.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates offer a different method for describing points on a plane compared to the traditional Cartesian coordinate system. Instead of using an ordered pair \(x, y\), polar coordinates use \(r\) and \(\theta\). What makes it unique is how it emphasizes distance and direction:

• \(r\): This is the radius or the direct distance of the point from the origin.
• \(\theta\): This represents the angle formed with the positive x-axis. It's typically measured in radians, but degrees are also used at times.

Imagine a point on a piece of paper that you could reach by measuring a certain distance from its center at a specified angle. This perspective is particularly useful in situations where symmetry and rotation play a central role, such as in circular motion or where curves return to a starting point, like spirals.

Coordinate Conversion

Converting between polar and Cartesian coordinates may sound tricky initially, but it's based on some simple trigonometric relationships. The process involves linking the polar variables \(r\) and \(\theta\) with Cartesian variables \(x\) and \(y\), which define points on the Cartesian plane. Here's how the conversion works:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• Using Pythagoras’s theorem, \(r^2 = x^2 + y^2\)

One of the main reasons for using these conversions is to translate between the useful geometry of circular patterns or rotations in polar coordinates and the linear format of Cartesian equations. For instance, if we know \(r\) and \(\theta\), we can effortlessly find \(x\) and \(y\) using the formulas above.
This seamless conversion allows us to express polar curves in Cartesian format, making it simpler to integrate them into functions or equations that require Cartesian coordinates.

Curve Identification

Identifying curves involves translating their polar representations into Cartesian equations, which can make them easier to recognize and classify. In the process of converting from polar to Cartesian form, we aim to rediscover curves familiar in algebraic studies.

For instance, with an equation like \(r = 3 \sin \theta\), by converting it, we found an equivalent Cartesian form: \(x^2 + y^2 = 3y\). Upon rearranging, \(x^2 + y^2 - 3y = 0\) emerges, which is a circle shifted along the y-axis.
This specific manipulation helps reveal the type of curve, in this case, a circle with a center at \(0, \frac{3}{2}\) and a radius of \(\frac{3}{2}\). Curve identification is crucial for revealing the geometric nature or the symmetry of the original shape, which might not be immediately apparent from its polar form. This step is essential, especially in calculus or engineering tasks, where interpreting the shape can influence how we approach problems involving the curve.