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Chapter 6: Problem 5

The equation \(r^{2}=4 \sin 2 \theta\) represents a ________________.

Short Answer

Expert verified
The equation \(r^{2}=4 \sin 2 \theta\) represents a tilted lima莽on.

Step by step solution

01

Identify the polar equation form

Firstly, analyze the equation. When solved for \(r\), the given equation becomes \(r=\sqrt{4 \sin 2 \theta}\). This does not match any common polar form exactly.
02

Find an alteration that might simplify the form

Use the double-angle formula to rewrite the equation, \(r^{2}=4 \sin 2 \theta\), as \(r^2=2(1-\cos(2\theta))\).
03

Identify obtained form

Recognize the new equation as a tilted lima莽on. This is considering that \(r^2=2(1-\cos(2\theta))\) is a form of \(r=2(1- \cos \theta)\) whereby it is scaled by \(\sqrt{2}\) and titled by 蟺/4.

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Key Concepts

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

Understanding Polar Coordinates
Polar coordinates offer a different way of describing the position of a point in a plane than the traditional Cartesian coordinates. Instead of using an x-coordinate and a y-coordinate, polar coordinates use a radius and an angle to define the location of a point around a central point. The polar coordinate system is denoted as \( (r, \theta) \), where:
  • \( r \) is the radial distance from the origin (central point) to the point.
  • \( \theta \) is the angle in radians measured from the positive x-axis, usually in the counter-clockwise direction.
To convert between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\), you can use these formulas:
  • \( x = r \cos(\theta) \)
  • \( y = r \sin(\theta) \)
  • \( r = \sqrt{x^2 + y^2} \)
  • \( \theta = \tan^{-1}(\frac{y}{x}) \)
Polar coordinates are especially useful in scenarios dealing with circular paths or periodic functions, simplifying the representation and calculation of such problems.
Double-Angle Formulas in Trigonometry
Double-angle formulas are pivotal in trigonometry, allowing us to express trigonometric functions of double angles in terms of single angles. These formulas can simplify and transform complex trigonometric calculations.For angle \(2\theta\), the double-angle formulas are:
  • \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
  • \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta) \)
These formulas are incredibly useful when dealing with equations involving trigonometric functions in polar coordinates, such as the one in the exercise \(r^2 = 4 \sin 2\theta\). In the solution, the double-angle formula for sine helps transform and recognize standard forms, which aids in identifying the type of curve depicted by a polar equation.
Exploring Lima莽on Curves
A Lima莽on is a type of polar curve, named after the French word for "snail" due to its distinctive shape. Lima莽on curves are versatile, appearing in different forms like the cardioid or when they develop an inner loop, depending on the parameters of the equation.The general equation for a Lima莽on in polar coordinates is \( r = a + b\cos(\theta) \) or \( r = a + b\sin(\theta) \). These curves exhibit interesting features:
  • If \( |a| = |b| \), the Lima莽on becomes a cardioid.
  • If \( |a| > |b| \), the Lima莽on has no inner loop.
  • If \( |a| < |b| \), the Lima莽on has an inner loop.
In the provided exercise, our equation \(r^2 = 2(1 - \cos(2\theta))\) hints towards a Lima莽on form when compared to \(r = 2(1 - \cos \theta)\), which has been tilted and scaled appropriately, illustrating how versatile and intricate polar equations can be.

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