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Chapter 7: Problem 97

For each rectangular equation, write an equivalent polar equation. $$y=2 x-1$$

Short Answer

Expert verified
The polar equation is \( r = \frac{-1}{ \sin(\theta) - 2 \cos(\theta)} \)

Step by step solution

01

Recall the Relationship Between Rectangular and Polar Coordinates

In polar coordinates, any point \(x, y\) in rectangular coordinates can be represented as \(r, \theta\). The relationships are as follows: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]
02

Substitute the Polar Coordinate Equivalents

In the given equation \( y = 2x - 1 \), substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). The equation becomes: \[ r \sin(\theta) = 2 \cdot r \cos(\theta) - 1\]
03

Simplify and Solve for r

Group the terms to isolate \(r\): \[ r \sin(\theta) = 2r \cos(\theta) - 1 \] \[ r \sin(\theta) - 2r \cos(\theta) = -1 \] Factor out \(r\) on the left side: \[ r ( \sin(\theta) - 2 \cos(\theta)) = -1 \] Solve for \(r\): \[ r = \frac{-1}{\sin(\theta) - 2 \cos(\theta)} \]

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Key Concepts

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

polar coordinates
Polar coordinates provide a different way to represent a point in a plane. Instead of using Cartesian coordinates \( (x, y) \), we use \( (r, \theta) \). Here, \( r \) is the distance from the origin to the point, and \( \theta \) is the angle measured from the positive x-axis to the line segment connecting the origin to the point.

The polar coordinates system is particularly useful for problems involving symmetry around a point. For example, when dealing with circular or spiral patterns, polar coordinates can make the math simpler and more intuitive.
rectangular coordinates
Rectangular coordinates, or Cartesian coordinates, represent points in a 2D plane using two values: \( x \) (the horizontal position) and \( y \) (the vertical position). These coordinates are extremely useful for arithmetic-based geometry and algebra.

For instance, plots on graph paper commonly utilize rectangular coordinates. The simplicity in determining distances and angles directly is where this system shines.

In equations, it鈥檚 straightforward to relate variables using familiar operations like addition and multiplication.
coordinate transformation
Coordinate transformation is the process of converting from one coordinate system to another. This is done using specific mathematical relationships.

For instance, to go from rectangular to polar coordinates, you use: \[ x = r \cos(\theta) \] and \[ y = r \sin(\theta) \]

Conversely, to convert from polar to rectangular, the equations are: \[ r = \sqrt{x^2 + y^2} \] and \[ \theta = \tan^{-1}(\frac{y}{x}) \]

This process of converting coordinates can be essential in geometry and physics, where different viewpoints simplify different aspects of a problem.
trigonometric functions
Trigonometric functions relate the angles of a triangle to the lengths of its sides. Key trigonometric functions include \(\sin\theta\), which gives the ratio of the opposite side to the hypotenuse, and \(\cos\theta\), the ratio of the adjacent side to the hypotenuse.

These functions are crucial when working with polar coordinates. For instance, in the transformation process, \( \sin \theta \) and \( \cos \theta \) help relate the angle \( \theta \) and distance \( r \) to traditional Cartesian coordinates \((x, y)\).

Understanding these trigonometric relationships is invaluable for converting equations and for many applications in calculus and physics.

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Most popular questions from this chapter

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