Problem 78 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ y=-3 $$

Step-by-Step Solution

Answer

\( r \sin \theta = -3 \)

Step 1: Identify the given equation in rectangular coordinates

The given equation is in rectangular coordinates: \[ y = -3 \]

Step 2: Write the rectangular coordinates in terms of polar coordinates

Recall the relationships between rectangular and polar coordinates: \[ x = r \, \cos \theta \]\[ y = r \, \sin \theta \]

Step 3: Substitute the polar coordinate relationship into the equation

We replace \( y \) with \( r \, \sin \theta \) from the polar coordinate system: \[ r \, \sin \theta = -3 \]

Step 4: Rewrite the polar equation

The equation \( y = -3 \) in polar coordinates is written as: \[ r \, \sin \theta = -3 \]

Key Concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a standard way of representing points in a plane using a pair of numerical values: the x-coordinate and the y-coordinate. These coordinates are written as \(x, y\). The x-coordinate represents the horizontal distance from the origin (0, 0) along the x-axis, while the y-coordinate represents the vertical distance from the origin along the y-axis. Rectangular coordinates are particularly useful in fields like graphing functions and plotting points in 2D space.
For example, in the given exercise, the equation \(y = -3\) specifies that all points have a y-coordinate of -3, meaning they lie on a horizontal line that is 3 units below the x-axis.

coordinate transformation

Coordinate transformation refers to the process of converting coordinates from one coordinate system to another. In this case, we are moving from rectangular coordinates \(x, y\) to polar coordinates \(r, \theta\). This is useful in many mathematical applications.
The relationships between the systems are:

• \( x = r \, \cos \theta \)
• \( y = r \, \sin \theta \)

To convert, you substitute these expressions into the given rectangular coordinate equation. In our example, substituting \( y = r \, \sin \, \theta \) into \( y = -3 \) gives us the equation \( r \, \sin \, \theta = -3 \). This way, we've transformed the rectangular equation \( y = -3 \) into its corresponding polar form.

trigonometric identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variable involved. When converting between rectangular and polar coordinates, certain trigonometric relationships help simplify the process. The primary identities used are:

• \( \sin \theta \) - opposite over hypotenuse in a right triangle
• \( \cos \theta \) - adjacent over hypotenuse

In our exercise, we replaced y with \( r \, \sin \, \theta \) using the identity because \( y = r \, \sin \theta \), which directly connects the vertical distance in rectangular coordinates to polar coordinates. Recognizing and using these identities can aid in making transformations between different coordinate systems much easier.