Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=4 \sin \theta $$

Short answer

The rectangular form is \(x^2 + (y - 2)^2 = 4\), a circle with center \((0, 2)\) and radius 2.

Step-by-step solution

Understanding Polar Coordinates

The given equation is in polar form, where each point on the plane is defined by a radius \(r\) and an angle \(\theta\). Our goal is to convert this into a rectangular coordinate system \((x, y)\).

Using Polar to Rectangular Conversion Formulas

To convert the polar equation \(r = 4 \sin \theta\) into rectangular form, recall that the conversions from polar to rectangular coordinates are: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, note that \(r^2 = x^2 + y^2\) and \(y = r \sin \theta\).

Substituting the Sin Conversion Equation

Since \(y = r \sin \theta\), we use \(r = 4 \sin \theta\) to get \(y = r \sin \theta = 4\sin \theta = 4\cdot\frac{y}{r}\).

Isolating r

We know from the polar equation that \(r = 4 \sin \theta\), which can be written as \(r = \frac{4y}{r}\). Multiply both sides by \(r\) to clear the fraction: \(r^2 = 4y\).

Using Radius to Rectangular Coordinate Conversion

Recall that \(r^2 = x^2 + y^2\). Substitute this into the equation found previously: \(x^2 + y^2 = 4y\).

Completing the Square for y

Rearrange the equation: \(x^2 + y^2 - 4y = 0\). To complete the square for the \(y\) terms, write it as \(x^2 + (y - 2)^2 - 4 = 0\). Simplifying, we get \(x^2 + (y - 2)^2 = 4\).

Identifying the Shape

The equation \(x^2 + (y - 2)^2 = 4\) represents a circle with center \((0, 2)\) and radius \(2\).

Sketching the Graph

Draw a circle centered at \((0, 2)\) with a radius of 2 units on the plane. This graph represents the converted rectangular form of the given polar equation.

Key concepts

Polar Coordinates

Polar coordinates define a point in a plane using a radius and an angle. It's like giving directions using distance and degrees. Polar coordinates consist of:

• Radius \( r \): the length from the origin (the center of the graph) to the point.
• Angle \( \theta \): the angle measured from the positive x-axis to the line connecting the origin and the point.

This system is particularly useful in situations involving circles and spirals. In our exercise, the equation \( r = 4 \sin \theta \) illustrates a relationship between the radius \( r \) and the angle \( \theta \). The angle determines the position on the plane, while the equation describes how far that point lies from the origin. It's flexible and can be more intuitive than rectangular coordinates when dealing with circular paths.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use a grid with x and y axes to define a point's position. Visualize it as a map with horizontal and vertical streets intersecting at right angles. Each point is defined as:

• x-coordinate \( x \): horizontal position from the y-axis.
• y-coordinate \( y \): vertical position from the x-axis.

To convert from polar to rectangular coordinates, use these relationships:

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

In our solution, we've converted \( r = 4 \sin \theta \) into a rectangular equation form \( x^2 + (y - 2)^2 = 4 \), which represents a circle with its center shifted along the vertical axis. With these simple formulas, translating points between systems becomes manageable and opens up new perspectives for graph interpretation.

Graphing Equations

Graphing equations involve representing mathematical relations visually, often aiding in understanding their behavior. For both polar and rectangular coordinates, interpreting graphs can be quite distinct. Here’s how you can approach graphing the equation from our solution:- Consider the circle centered at . It is represented as \( x^2 + (y - 2)^2 = 4 \), showing a circle with a center at \( (0, 2) \) and a radius of \( 2 \).- Use straightforward tools, like compasses or digital graphing software, for precise sketches.

- Begin with a grid layout and plot the circle accordingly:

• Locate the center of the circle on the grid at the coordinates \((0, 2)\).
• Measure and mark the radius by moving 2 units away from the center in each cardinal direction (up, down, left, right).

This graphical representation is the bridge between converting polar equations to rectangular form and visualizing them. Providing a clear, visual aspect of mathematical relationships can help clarify complicated solutions, making abstract concepts tangible.