Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=-4 \sin \theta $$

Short answer

The rectangular equation is \( y = -4 \). It represents a horizontal line passing through the point where y is -4.

Step-by-step solution

Use Polar to Rectangular Coordinate Transformations

The given polar equation is given as \( r = -4 \, \sin \theta \). Use the polar to rectangular conversions: \( x = r \, \cos \theta \) and \( y = r \, \sin \theta \). Additionally, remember that \( r^2 = x^2 + y^2 \).

Express Polar Equation Using \( y = r \, \sin \theta \)

From \( y = r \, \sin \theta \), we can substitute \( y \) in place of \( r \, \sin \theta \). The given equation is \( r = -4 \, \sin \theta \). Thus, \( y = -4 \).

Formulate The Rectangular Equation

Since there is no \( x \)-term or \( r \)-term left in the equation, it is evident that the rectangular equivalent of the given polar equation is simply \( y = -4 \).

Identify and Graph the Rectangular Equation

\( y = -4 \) represents a horizontal line in the rectangular coordinate system. This line is parallel to the x-axis and goes through the point where y is -4. To graph this line, simply draw a straight horizontal line that intersects the y-axis at -4.

Key concepts

polar coordinates

Polar coordinates are a way to represent points in the plane using a distance and an angle. In this system, each point is determined by the distance (radius) from a reference point (usually the origin) and an angle measured from a reference direction (usually the positive x-axis).
Commonly used notations for polar coordinates are \(r, \theta\). For example, \( r = 5, \theta = \frac{\pi}{3} \) means the point lies 5 units away from the origin and forms an angle of \(\frac{\pi}{3}\) radians with the x-axis.
Remember, angles can be expressed in degrees or radians, with 360 degrees equivalent to 2π radians.

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points using two perpendicular lines (axes). These coordinates are given as ordered pairs \(x, y\), where:

• \(x\): represents the horizontal distance from the origin.
• \(y\): represents the vertical distance from the origin.

For example, the point \(3, -2\) is 3 units to the right of the origin and 2 units down.
Rectangular coordinates are often more straightforward for visualizing linear relationships and distances compared to polar coordinates.

equation transformation

Equation transformation involves converting an equation from polar coordinates to rectangular coordinates, or vice versa. To accomplish this, you use specific formulas:

• Converting from polar to rectangular: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Converting from rectangular to polar: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).

When you have an equation like \(r = -4 \sin \theta\), use \(y = r \sin \theta\). Since \( y = -4 \), this means the equation in rectangular coordinates doesn't depend on x, forming a line parallel to the x-axis at \(y = -4\).

graphing equations

Graphing equations involves plotting points or lines on a coordinate system to visually represent the relationship defined by the equation.
For the equation \(y = -4\), this represents a horizontal line. Here's how you can graph based on the conversion result:

• Identify the y-intercept: Here it is \(-4\). This is where the line crosses the y-axis.
• Draw a straight horizontal line: Extend this line horizontally across all x-values but maintain that it passes through the y-axis at \(-4\).

This line will effectively be parallel to the x-axis, demonstrating simpler relationships in rectangular coordinates compared to polar coordinates.