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Magazine Chapter 8: Problem 42
Convert the equation to polar form. $$x^{2}+y^{2}=9$$
Short Answer
Expert verified
The polar form is \( r = 3 \).
Step by step solution
01
Identify the Polar Coordinates
First, we need to understand the relation between Cartesian coordinates \(x, y\) and polar coordinates \(r, \theta\). In polar form, \(x = r \cos\theta\) and \(y = r \sin\theta\). Also, \(x^2 + y^2 = r^2 \).
02
Substitute the Polar Definitions
Using the polar coordinate definitions, substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the equation \(x^{2}+y^{2}=9\). This gives \[ (r \cos\theta)^2 + (r \sin\theta)^2 = 9. \]
03
Simplify the Equation
Simplify \[ (r \cos\theta)^2 + (r \sin\theta)^2 = r^2 (\cos^2\theta + \sin^2\theta). \] Using the trigonometric identity \(\cos^2\theta + \sin^2\theta = 1\), the equation simplifies to \(r^2 = 9\).
04
Solve for r
To solve for \(r\), take the square root of both sides: \[ r = 3 \. \]
05
Write the Polar Form
The polar form of the given equation \(x^2 + y^2 = 9\) is \(r = 3\). This represents a circle of radius 3 centered at the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cartesian Coordinates
Understanding Cartesian coordinates is essential when dealing with equations involving x and y. Cartesian coordinates use two values, x and y, to denote the position of a point in a two-dimensional plane. Think of it like using street addresses to identify a location.
- X-coordinate: Represents the point's horizontal position from the origin (0,0).
- Y-coordinate: Represents the point's vertical position from the origin.
- Origin: The center of the coordinate system where x = 0 and y = 0.
Trigonometric Identities
Trigonometric identities play a crucial role in converting equations from Cartesian coordinates to polar coordinates. These identities are essentially equations that are true for all angles. The most common one to remember here is:
- Pythagorean Identity: \( \cos^2\theta + \sin^2\theta = 1 \)
Equation Conversion
Equation conversion is the process of transforming an equation from one form into another. In our exercise, we convert the equation \(x^2 + y^2 = 9\) from a Cartesian form to a polar form.
Here's how the conversion works:
Here's how the conversion works:
- Start with the equation written in Cartesian form.
- Use the relationships between Cartesian and polar coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\).
- Substitute these expressions into the equation, which transforms it into \((r \cos \theta)^2 + (r \sin \theta)^2 = 9\).
- Simplify the equation using trigonometric identities like \(\cos^2 \theta + \sin^2 \theta = 1\) to get \(r^2 = 9\).
- Finally, solve for r to determine the polar equation, resulting in \(r = 3\).
Radius of a Circle
The radius of a circle is a fundamental concept in geometry, it's the distance from the center of the circle to any point on its circumference. In polar coordinates, solving an equation like \(x^2 + y^2 = 9\) involves finding this radius.
In our polar equation, \(r = 3\) specifically tells us:
In our polar equation, \(r = 3\) specifically tells us:
- This is the radius, showing each point on the circle is 3 units from the origin.
- The circle is perfectly centered at the origin (0,0).
