Question

Graph the curve defined by the function \(r=4 \sin \theta\). Identify the curve and rewrite the equation in rectangular coordinates.

Short answer

The curve is a circle centered at (0, 2) with radius 2, described by the rectangular equation \(x^2 + (y - 2)^2 = 4\).

Step-by-step solution

Identify the polar equation

The given polar equation is \( r = 4 \sin \theta \). This equation represents a sinusoidal relationship between the radial distance \( r \) and the angle \( \theta \).

Describe the concept of a polar graph

The graph of \( r = 4 \sin \theta \) is a circle in the polar coordinate system. Here, the circle will be symmetric about the vertical axis because of the sine function.

Determine key points of the polar graph

For \( \theta = 0 \), \( r = 0 \). For \( \theta = \frac{\pi}{2} \), \( r = 4 \). For \( \theta = \pi \), \( r = 0 \). For \( \theta = \frac{3\pi}{2} \), \( r = -4 \). Plot these points to see the circle.

Convert the polar equation to rectangular coordinates

To convert \( r = 4 \sin \theta \) to rectangular form, use the identities: \( x = r \cos \theta \), \( y = r \sin \theta \), \( r^2 = x^2 + y^2 \), and \( \sin \theta = \frac{y}{r} \). Thus, \( r = 4 \frac{y}{r} \) becomes \( r^2 = 4y \). Substitute \( r^2 = x^2 + y^2 \) to get \( x^2 + y^2 = 4y \).

Rearrange to standard form

Rearrange the equation \( x^2 + y^2 = 4y \) to find its standard form. Complete the square for the \( y \/ term: x^2 + (y - 2)^2 = 4 \). This indicates a circle with center at \((0, 2)\) and radius 2.

Key concepts

Polar Equations

A polar equation is a mathematical expression that represents relationships in terms of polar coordinates - defined by the radial distance \( r \) and the angular coordinate \( \theta \). The equation \( r = 4 \sin \theta \) exemplifies a polar equation by using sine to define the variation of \( r \) relative to \( \theta \). This type of equation is particularly useful for graphing patterns that are naturally circular or periodic.
Polar equations can describe many curves, but they are most commonly used for curves symmetric about the origin or a particular line, such as circles and spirals. In our example with \( r = 4 \sin \theta \), the curve turns out to be a circle with a diameter of 4.

• They can effectively describe shapes that don't align well with standard cartesian axes.
• Understanding the role of different trigonometric functions helps in visualizing their graphs.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use an \(x-y\) grid to specify locations on a plane through horizontal and vertical distances. This system involves an \(x\) value and a \(y\) value to define points.
In rectangular coordinates, the equation of a circle is typically written in the form \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. By converting polar equations into this form, it becomes easier to work with them in contexts needing Cartesian logic or algorithms.

• Efficient for linear and quadratic graphs.
• Provides a straightforward approach to mathematical modeling and calculations.

Conversion Between Coordinate Systems

To convert between polar and rectangular coordinates involves using certain identities and trigonometric relationships.
For the conversion from polar to rectangular coordinates, use these main equations:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \tan \theta = \frac{y}{x} \)

These transformations facilitate the conversion of polar curves like the one given, \( r = 4 \sin \theta \), into their rectangular counterparts such as \( x^2 + y^2 = 4y \).
In the example exercise, after recognizing the polar equation, we use \( r = 4 \frac{y}{r} \) and substitute \( r^2 = x^2 + y^2 \) to convert the curve into rectangular coordinates, resulting in the equation \( x^2 + y^2 = 4y \), which simplifies further to \((x^2 + (y-2)^2 = 4)\).

• Conversions require a sound understanding of trigonometric identities.
• They allow flexibility across different mathematical problems and applications.