Problem 22 - Algebra and Trigonometry

Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \sin \theta=-2 $$

Step-by-Step Solution

Answer

The equation in rectangular coordinates is \( y = -2 \), which is a horizontal line at \( y = -2 \).

Step 1: Rewrite the Polar Equation

Start by writing the given polar equation using trigonometric identities. The given polar equation is:\[ r \, \sin \theta = -2 \]

Step 2: Use the Polar to Rectangular Conversion

We know that the conversion from polar to rectangular coordinates for \( y \) is given by \( y = r \, \sin \theta \). Therefore, substitute \( y \) in place of \( r \, \sin \theta \):\[ y = -2 \]

Step 3: Identify and Examine the Rectangular Equation

The resulting rectangular equation is:\[ y = -2 \] This equation represents a horizontal line in the rectangular coordinate system.

Step 4: Graph the Equation

Graph the rectangular equation by drawing a horizontal line that crosses the y-axis at \( -2 \). This means that for all values of \( x \), the value of \( y \) will be \( -2 \).

Key Concepts

Trigonometric Identities

To transform polar equations to rectangular coordinates, we often rely on trigonometric identities. These identities help us convert between the two forms using familiar relationships. For example:
- \( x = r \, \cos \theta \)
- \( y = r \, \sin \theta \)
In the given problem, the identity \( y = r \, \sin \theta \) is crucial. This identity allows us to switch from polar coordinates, which depend on the radius (r) and angle (\theta), to rectangular coordinates, which are based on x and y values.

Graphing Equations

Understanding how to graph equations is essential. Converting from polar to rectangular coordinates involves more than just changing the variables- it also involves understanding how these equations look when graphed.
For instance, in the exercise, the rectangular equation we obtain is \( y = -2 \). This represents a horizontal line.
A few important points while graphing:
- **Horizontal Line**: A line parallel to the x-axis, indicating that y remains constant.
- **Vertical Line**: A line parallel to the y-axis, indicating that x remains constant.
Graphing equations in the rectangular system is often about identifying these patterns and translating them correctly.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, implement a grid system to describe the position of points. Unlike polar coordinates, which use a radius and angle, rectangular coordinates are straightforward with x and y values. In this exercise:
- We start with a polar equation \( r \, \sin \theta = -2 \) and convert it to \( y = -2 \).
This means, regardless of the x value, y will always be \ -2 \.
Why this is useful:
- It's easier to plot points in a straightforward grid system.
- Allowing direct visualization of distance and angles can sometimes be more intuitive than polar coordinates.

Horizontal Line

A horizontal line in the rectangular coordinate system is represented by \( y \) = a constant value. This line tells us that no matter where you are along the x-axis, the y value does not change. In the exercise, the line \( y = -2 \):
- Passes through the y-axis at \ y = -2 \
- Extends infinitely to the left and right
- Shows that for every \ x \ value, y remains at \ -2 \.
Such lines are simple to graph and understand, providing a clear visual representation of equations like the one given in the exercise.