Problem 41 Find an equation in $x$ and \(... [FREE SOLUTION]

Chapter 10: Problem 41

Find an equation in $x$ and $y$ that has the same graph as the polar equation. Use it to help sketch the graph In an $r \theta$ -plane. $$r=8 \sin \theta-2 \cos \theta$$

Short Answer

Expert verified

The Cartesian form is $(x+1)^2 + (y-4)^2 = 17$; it represents a circle with a center at (-1, 4) and radius $ \sqrt{17} $.

Step by step solution

Convert Polar to Cartesian Coordinates

In polar coordinates, we have $ r = 8 \sin \theta - 2 \cos \theta $. Recall the relations between polar and Cartesian coordinates: $ x = r \cos \theta $ and $ y = r \sin \theta $. We also have $ x = r \cos \theta $ and $ y = r \sin \theta $ which gives $ r^2 = x^2 + y^2 $. Using these relationships, let's express $ r $ in terms of $ x $ and $ y $.

Express Trigonometric Functions in Terms of Coordinates

First, express $ \sin \theta $ and $ \cos \theta $ in terms of $ x $, $ y $, and $ r $:- $ \sin \theta = \frac{y}{r} $- $ \cos \theta = \frac{x}{r} $Substitute these into the original equation: $ r = 8 \left( \frac{y}{r} \right) - 2 \left( \frac{x}{r} \right) $This simplifies to:$ r^2 = 8y - 2x $

Rearrange to Form a Cartesian Equation

We know from polar coordinates that $ r^2 = x^2 + y^2 $. Equating this to the expression we found earlier, $ r^2 = 8y - 2x $, gives us a simplification in Cartesian form:$ x^2 + y^2 = 8y - 2x $Rearrange to get:$ x^2 + 2x + y^2 - 8y = 0 $

Complete the Square

We'll complete the square for both $ x $ and $ y $.For $ x^2 + 2x $:$ x^2 + 2x $ becomes $ (x+1)^2 - 1 $.For $ y^2 - 8y $:$ y^2 - 8y $ becomes $ (y-4)^2 - 16 $.The equation thus becomes:$(x + 1)^2 - 1 + (y - 4)^2 - 16 = 0 $.

Simplify and identify the graph

Combine constants to simplify the expression:$(x + 1)^2 + (y - 4)^2 = 17 $ This represents a circle with center at (-1, 4) and radius $ \sqrt{17} $. On the polar graph, this circle is centered around the point (8, $-2$) due to the transformations from the origin.

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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 system for representing points in a plane using a radius and an angle. Instead of describing a point by its horizontal and vertical distances from an origin (like Cartesian coordinates), polar coordinates describe a point by how far it is from the origin and the angle from a reference direction.

The distance from the origin to the point is represented by 'r'.
The angle from the positive x-axis is represented by '胃' (theta).

This system is particularly useful for circular or rotational systems because it naturally fits the radial and angular orientation of a circle.

Cartesian Coordinates

Cartesian coordinates use a grid system to specify the position of points in a plane using two numbers. These numbers indicate the distance from two perpendicular intersecting lines known as the x-axis and y-axis.

'x' represents the horizontal distance from the y-axis.
'y' represents the vertical distance from the x-axis.

This method is ideal for regular grids and rectangular objects. The conversion from polar to Cartesian involves using the relationships: $x = r \cos \theta$ and $y = r \sin \theta$. These formulas are derived from the right triangle relationships based on trigonometry.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square plus a constant. This makes it easier to analyze and graph equations, especially conics like circles and parabolas. Here is a quick guide:

For a quadratic form like $x^2 + bx$, add and subtract $(\frac{b}{2})^2$ to form a perfect square trinomial: $(x + \frac{b}{2})^2$.
This process reveals the center and radius of a circle in a graph.

In our example, completing the square revealed the center of the circle at (-1, 4) and identified it as the equation of a circle.

Graphing Equations

Graphing equations involves creating a visual representation of algebraic equations, giving us valuable insights into their properties and behaviors. The Cartesian coordinate system is pivotal in this process.

Identify the type of equation: linear, quadratic, or in this case, circular.
Use completed square forms to easily find key attributes, like a circle's center and radius.

Given the example transition from polar to Cartesian, the graph of the equation $(x + 1)^2 + (y - 4)^2 = 17$ enabled us to accurately plot a circle with a specific center and radius, thus visualizing the spatial relationships dictated by the equation.

91影视

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph In an \(r \theta\) -plane. $$r=8 \sin \theta-2 \cos \theta$$

Short Answer

Step by step solution

Convert Polar to Cartesian Coordinates

Express Trigonometric Functions in Terms of Coordinates

Rearrange to Form a Cartesian Equation

Complete the Square

Simplify and identify the graph

Key Concepts

Polar Coordinates

Cartesian Coordinates

Completing the Square

Graphing Equations

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