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Chapter 6: Problem 84

For each rectangular equation, write an equivalent polar equation. $$ y=-3 x+5 $$

Short Answer

Expert verified
The polar equation equivalent to \(y = -3x + 5\) is \[ r = \frac{5}{\sin \theta + 3 \cos \theta} \]

Step by step solution

01

Recall the Rectangular and Polar Coordinates Relationship

In polar coordinates, the relationship between rectangular coordinates \(x, y\) and polar coordinates \(r, \theta\) are defined as: \[ x = r \cos \theta \] \[ y = r \sin \theta \]
02

Substitute Rectangular Coordinates

Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the given rectangular equation \(y = -3x + 5\). The equation becomes: \[ r \sin \theta = -3 r \cos \theta + 5 \]
03

Simplify the Equation

Factor out \(r\) from the terms on the left side of the equation: \[ r \sin \theta + 3 r \cos \theta = 5 \] \[ r (\sin \theta + 3 \cos \theta) = 5 \]
04

Solve for \(r\)

Isolate \(r\) by dividing both sides of the equation by \((\sin \theta + 3 \cos \theta)\): \[ r = \frac{5}{\sin \theta + 3 \cos \theta} \]

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Key Concepts

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

rectangular coordinates
Rectangular coordinates are a common way to describe the position of a point in a plane. They consist of two values, typically denoted as \(x\) and \(y\). These values indicate how far the point is from the origin along the horizontal (x-axis) and vertical (y-axis) directions.

For instance, the point \( (3, 4) \) means it is 3 units to the right of the origin and 4 units up. Rectangular coordinates are often used in algebra and geometry to plot graphs, solve equations, and more. Understanding the basics of these coordinates is essential for converting between different coordinate systems.
polar coordinates
Polar coordinates are another system for describing points in a plane, but they use a different approach. Instead of horizontal and vertical distances, polar coordinates describe a point's location using a radius and an angle.
  • The radius \( r \) is the distance from the point to the origin.
  • The angle \( \theta \) is the angle formed with the positive x-axis.

For example, the point \( (r, \theta) = (5, \frac{\theta}{4}) \) means it is 5 units away from the origin at an angle of \( \frac{\theta}{4} \) radians (or 45 degrees). This system is particularly useful in cases involving circular or rotational symmetry.
coordinate transformation
Coordinate transformation is the process of converting one coordinate system to another. In this context, we are converting rectangular coordinates to polar coordinates. The key relationships you'll use are:
  • \(x = r \cos \theta \)
  • \(y = r \sin \theta \)

To transform the given equation \(y = -3x + 5 \) into polar form:
  • Replace \(x \) with \(r \cos \theta \)
  • Replace \(y \) with \(r \sin \theta \)
Now, the equation becomes \( r \sin \theta = -3 r \cos \theta + 5 \). This is the beginning of the transformation process.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for a particular variable. Here, we need to solve for \(r \) after substituting the rectangular coordinates with polar ones.

The given equation \( r \sin \theta = -3 r \cos \theta + 5 \) is simplified by factoring out \(r \) from the terms:
\[ r (\sin \theta + 3 \cos \theta) = 5 \]
To isolate \(r \), divide both sides of the equation by \( (\sin \theta + 3 \cos \theta) \):
\[ r = \frac{5}{\sin \theta + 3 \cos \theta} \]

This final step gives the polar form of the original rectangular equation. Mastering algebraic manipulation is crucial for solving and transforming various equations in mathematics.

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Most popular questions from this chapter

For each rectangular equation, write an equivalent polar equation. $$ 2 x+3 y=6 $$

Perform the indicated operation: $$ 2(\cos (\pi / 6)+i \sin (\pi / 6)) \cdot 3(\cos (\pi / 3)+i \sin (\pi / 3)) $$ Write the result in \(a+b i\) form.

Write each complex number in trigonometric form, using degree measure for the argument. \(-3+3 i\)

For each polar equation, write an equivalent rectangular equation. $$ \theta=0 $$

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