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Chapter 6: Problem 80

Convert the rectangular equation to polar form. Assume \(a>0\). \(x^{2}+y^{2}=9 a^{2}\)

Short Answer

Expert verified
The polar form of the given rectangular equation \(x^{2}+y^{2}=9a^{2}\) is \(r = 3a\).

Step by step solution

01

Understanding the conversions

Recall that the conversion from rectangular to polar coordinates involves the equations \(r = \sqrt{x^{2} + y^{2}}\) and \(\tan(胃) = y / x\). For conversion the other way around, \(x = r \cos(胃)\) and \(y = r \sin(胃)\).
02

Substituting rectangular coordinates with polar coordinates

You can solve the problem now by substituting the rectangular coordinates (x, y) in the given equation with their polar forms (using \(x = r \cos(胃)\) and \(y = r \sin(胃)\)). This gives you \((r \cos{胃})^2 + (r \sin{胃})^2 = 9a^2\).
03

Simplify the Equation

You can further simplify the equation to \(r^{2} \cos^{2}{胃} + r^{2} \sin^{2}{胃} = 9a^2\). Now, you can factor \(r^{2}\) out, so you have \(r^{2}(\cos^{2}{胃} + \sin^{2}{胃}) = 9a^{2}\).
04

Apply Trig Identity

Recall that the trigonometric ideantity \(\cos^2胃 + \sin^2胃 = 1\). Use this identity, to simplify your equation further, yielding \(r^{2} = 9a^{2}\).
05

Solve for r

The final step is to solve for r. This can be done by taking the square root of both sides, which results in \(r = 3a\), which is the polar form of the given equation.

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, serve as a foundation for understanding various geometric problems. In this coordinate system, any point on a plane is determined using two coordinates:
  • **x-coordinate**: Represents the horizontal distance from the origin.
  • **y-coordinate**: Indicates the vertical distance from the origin.
These two values give a direct location by visualization on a grid system, commonly known for its familiar layout of (x, y). The rectangular equation given in the exercise, \(x^2 + y^2 = 9a^2\), represents a circle centered at the origin. Understanding this form is crucial when working on conversion to other types of coordinate systems such as polar coordinates.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They play a vital role in simplifying expressions and solving equations. The identity \(\cos^2{\theta} + \sin^2{\theta} = 1\) is one of the Pythagorean identities and is fundamental in converting rectangular coordinates to polar coordinates.
  • **Cosine function** measures the adjacent side over hypothenuse in a right triangle, which relates to the x-axis.
  • **Sine function** measures the opposite side over hypothenuse, relating to the y-axis.
In the exercise, this identity simplifies the equation obtained in polar form, helping reach the final form \(r^2 = 9a^2\). This simplification shows how trigonometric identities enable solving complex problems by reducing the terms involved.
Coordinate Conversion
Coordinate conversion between rectangular and polar coordinates is a common task. Each point in a 2D plane can be represented in both forms, thus understanding how to convert one to another is essential.
  • **From rectangular to polar coordinates:** The equations \(r = \sqrt{x^2 + y^2}\) and \(\tan(\theta) = \frac{y}{x}\) help in finding the corresponding polar coordinates \((r, \theta)\).
  • **From polar to rectangular coordinates:** Using \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\), you can retrieve the Cartesian statements.
In the example exercise, substituting \((x, y)\) with \((r \cos{\theta}, r \sin{\theta})\) allowed the transformation from the Cartesian circle equation \(x^2 + y^2 = 9a^2\) into a simple polar form \(r = 3a\). These conversions are essential for visualizing and solving geometric problems in differing contexts.

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Most popular questions from this chapter

Find the standard form of the equation of the ellipse with the given characteristics. Center: (2,-1)\(;\) vertex: \(\left(2, \frac{1}{2}\right) ;\) minor axis of length 2

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