Problem 50 Replace the polar equations with... [FREE SOLUTION]

Chapter 10: Problem 50

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=2 \cos \theta-\sin \theta$$

Short Answer

Expert verified

The Cartesian equation is a circle: $x^2 + y^2 - 2x + y = 0$.

Step by step solution

Identify Polar to Cartesian Conversion Formulas

To convert from polar to Cartesian coordinates, use the relationships: $ x = r \cos \theta $, $ y = r \sin \theta $, $ r^2 = x^2 + y^2 $, and $ \tan \theta = \frac{y}{x} $. In this case, we will use the first two relationships.

Express Trigonometric Functions

Identify the expressions for $ \cos \theta $ and $ \sin \theta $ using the conversion formulas: $ \cos \theta = \frac{x}{r} $ and $ \sin \theta = \frac{y}{r} $.

Substitute Trigonometric Expressions

Substitute $ \cos \theta = \frac{x}{r} $ and $ \sin \theta = \frac{y}{r} $ into the polar equation: $ r = 2 \frac{x}{r} - \frac{y}{r} $.

Multiply Through by r

To eliminate the fractions, multiply every term by $ r $: $ r^2 = 2x - y $.

Replace r^2 with x^2 + y^2

Use the identity $ r^2 = x^2 + y^2 $ to substitute on the left side: $ x^2 + y^2 = 2x - y $.

Rearrange into Standard Form

Rearrange the equation to standard form by moving all terms to one side: $ x^2 + y^2 - 2x + y = 0 $.

Identify the Graph

Identify the type of graph. This equation represents a circle in the Cartesian plane with center $(1, -\frac{1}{2})$ and radius $ \frac{\sqrt{5}}{2} $. Complete the square for $ x $ and $ y $ to confirm.

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Key Concepts

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

Cartesian Coordinates

Cartesian coordinates are used to identify a point on a plane using a pair of numerical values. These coordinates, often denoted as $(x, y)$, represent a linear system for organizing points in two-dimensional space.

Each point is described by an $x$-value (horizontal position) and a $y$-value (vertical position). This system allows for easy plotting and understanding geometric shapes and functions.

The $x$-coordinate tells how far along the point is on the horizontal axis.
The $y$-coordinate indicates the height of the point on the vertical axis.

This understanding makes analyzing and converting other forms like polar coordinates more straightforward when dealing with complex graphs or equations, such as transforming a polar equation $r = 2 \cos \theta - \sin \theta$ into its Cartesian counterpart.

Polar Equations

Polar equations involve expressions based on polar coordinates, typically describing the relationship between the distance from a point to the origin (denoted as $r$) and the angle from the positive $x$-axis to the point ($\theta$).

These equations utilize trigonometric functions $\cos$ and $\sin$, making them ideal for circular and spiral shapes.
The conversion from polar to Cartesian forms involves trigonometric identities like $x = r \cos \theta$ and $y = r \sin \theta$.

For instance, given a polar equation $r = 2 \cos \theta - \sin \theta$, we utilize these conversion formulas to translate it into a more familiar form, allowing for easier geometric interpretation on the Cartesian plane.

Graph Identification

Graph identification involves recognizing and classifying the type of graph represented by an equation. After transforming a polar equation to its Cartesian form, the next step is to determine the shape and characteristics of the graph.

For the equation $x^2 + y^2 - 2x + y = 0$, identifying patterns or standard forms can reveal if it's a circle, ellipse, parabola, etc.
Completing the square is a common technique to re-organize equations into recognizable forms.

In our example, by rewriting the Cartesian equation $x^2 + y^2 - 2x + y = 0$ with completed squares, we find it represents a circle with center at $(1, -\frac{1}{2})$ and radius $\frac{\sqrt{5}}{2}$. This step is crucial for understanding and visualizing how algebraic manipulations correspond to geometric forms on the Cartesian plane.

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91影视

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=2 \cos \theta-\sin \theta$$

Short Answer

Step by step solution

Identify Polar to Cartesian Conversion Formulas

Express Trigonometric Functions

Substitute Trigonometric Expressions

Multiply Through by r

Replace r^2 with x^2 + y^2

Rearrange into Standard Form

Identify the Graph

Key Concepts

Cartesian Coordinates

Polar Equations

Graph Identification

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