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Chapter 6: Problem 63

For each rectangular equation, write an equivalent polar equation. $$ y=3 $$

Short Answer

Expert verified
The equivalent polar equation is \[ r \sin(\theta) = 3 \].

Step by step solution

01

Understand the rectangular equation

The given equation is a horizontal line in the rectangular (Cartesian) coordinate system: \[ y = 3 \].
02

Recall the relationship between rectangular and polar coordinates

In polar coordinates, \[ x = r \cos(\theta) \] and \[ y = r \sin(\theta) \].
03

Substitute the polar coordinates into the equation

Since \[ y = 3 \], substitute using \[ y = r \sin(\theta) \]:\[ r \sin(\theta) = 3 \].
04

Write the equivalent polar equation

The polar coordinate equivalent of the rectangular equation \[ y = 3 \] is \[ r \sin(\theta) = 3 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polar coordinates
Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of the usual x and y coordinates, in polar coordinates we use:
  • \(r\): The distance from the point to the origin (0,0).
  • \(\theta\): The angle measured from the positive x-axis to the point.
For example, if \(r = 5\) and \(\theta = 45^\text{掳}\), then our point is 5 units away from the origin at a 45掳 angle from the positive x-axis.
rectangular coordinates
Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using x and y values. Each point is specified by how far it is along the x (horizontal) axis and how far it is along the y (vertical) axis.
  • \(x\): The horizontal distance from the origin (left or right on the x-axis).
  • \(y\): The vertical distance from the origin (up or down on the y-axis).
The point (3, 4) means moving three units to the right and four units up from the origin. It's very similar to plotting points on graph paper.
coordinate transformation
Coordinate transformation is the process of converting coordinates from one system to another. For instance, transforming rectangular coordinates to polar coordinates involves using specific formulas:
  • \(x = r \, \text{cos} \, \theta\)
  • \(y = r \, \text{sin} \, \theta\)
Conversely, converting polar coordinates back to rectangular coordinates uses:
  • \(r = \, \text{sqrt}\ (x^2 + y^2)\)
  • \(\theta = \, \text{atan2}\ (y, x)\)
This way, a point can be represented in either the polar or rectangular system, relying on the relationships mentioned.
trigonometric substitution
Trigonometric substitution is used in converting between rectangular and polar coordinates by replacing x and y with their trigonometric equivalents involving \(r\) and \(\theta\). For the given equation \(y = 3\), we use:
  • \(y = r \, \text{sin}\ (\theta)\)
Substituting this into our equation, we get
  • \(r \, \text{sin}\ (\theta) = 3\)
Thus, the rectangular equation \(y = 3\) is converted to the polar equation \(r \, \text{sin}\ (\theta) = 3\). This equation now expresses the same line in the polar coordinate system.

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Most popular questions from this chapter

Write each complex number in trigonometric form, using degree measure for the argument. \(4-4 i\)

Write \(4-4 i\) in trigonometric form, using degree measure for the argument.

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