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Magazine Chapter 11: Problem 38
Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r^{2} \sin 2 \theta=2$$
Short Answer
Expert verified
The equivalent Cartesian equation is \( xy = 1 \), which is a hyperbola.
Step by step solution
01
Understand the Polar Equation
The given polar equation is \( r^2 \sin 2\theta = 2 \). This is a polar equation involving the product of \( r^2 \) and the sine of double the angle \( \theta \). Our task is to convert this into a Cartesian form.
02
Use Polar to Cartesian Transformations
Recall the relationships between polar and Cartesian coordinates: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Moreover, \( \sin 2\theta = 2 \sin \theta \cos \theta \). Our goal is to replace \( r \) and \( \theta \) in the equation using \( x \) and \( y \).
03
Substitute the Double Angle Identity
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), substitute into the polar equation: \( r^2 (2 \sin \theta \cos \theta) = 2 \). Simplify to get \( 2r^2 \sin \theta \cos \theta = 2 \).
04
Solve for \( \sin \theta \) and \( \cos \theta \)
Divide both sides by 2 to get \( r^2 \sin \theta \cos \theta = 1 \). Now substitute \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \) into the equation.
05
Replace \( r \) with Cartesian Coordinates
Substitute \( \sin \theta \cos \theta = \frac{y}{r} \cdot \frac{x}{r} = \frac{xy}{r^2} \). Therefore, we can write \( r^2 \frac{xy}{r^2} = 1 \), which simplifies to \( xy = 1 \). This gives the Cartesian equation.
06
Identify the Graph
The equation \( xy = 1 \) is that of a hyperbola. It is a rectangular hyperbola centered at the origin with its asymptotes along the coordinate axes.
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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 unique way of representing points in a plane using a distance and an angle. Instead of traditional Cartesian coordinates which use an x and y axis, polar coordinates use:
- Radius ( "): The distance from the point to the origin.
- Angle ( "): The angle measured counterclockwise from the positive x-axis.
Cartesian Coordinates
Cartesian coordinates describe a point's position using two numbers, typically represented as
, the point's horizontal distance from the origin.
- x-coordinate (abscissa): It indicates the horizontal position.
- y-coordinate (ordinate): It indicates the vertical position.
Double Angle Identity
The double angle identity is a crucial trigonometric formula useful in simplifying expressions involving angles. It expresses trigonometric functions of double angles in terms of single angles, such as:
- heta = 2 heta = heta heta = heta heta) These identities enable conversion from polar expressions to Cartesian equations more easily, serving as a helpful bridge in many mathematical problems.
Hyperbola Graph
A hyperbola is a type of conic section, which forms when a plane intersects both halves of a double cone. The main features of a hyperbola include:
- Two disconnected curves called branches.
- Certain asymptotes that guide the curves' shape without ever crossing them.
