Chapter 10: Problem 91
Express \(r^{2}=\cos (2 \theta)\) in rectangular coordinates free of radicals.
Short Answer
Expert verified
\((x^{2} + y^{2})^2 = x^{2} - y^{2}\)
Step by step solution
01
Understand the given equation
The given equation is in polar coordinates: \(r^{2}=\text{cos}(2\theta)\). We need to express this equation in rectangular coordinates.
02
Recall polar to rectangular coordinate transformations
Recall the transformations from polar coordinates to rectangular coordinates: 1. \(x = r \text{cos}(\theta)\) 2. \(y = r \text{sin}(\theta)\) 3. \(r^{2} = x^{2} + y^{2}\) 4. \( \text{cos}(2\theta) = \text{cos}^{2}(\theta) - \text{sin}^{2}(\theta) \)
03
Substitute \(r^{2}\) in the given equation
From the transformations, substitute \(r^{2} = x^{2} + y^{2}\) into the given equation \(r^{2} = \text{cos}(2\theta)\): \[ x^{2} + y^{2} = \text{cos}(2\theta) \]
04
Express \(\text{cos}(2\theta)\) in terms of \(\text{cos}(\theta)\) and \(\text{sin}(\theta)\)
From trigonometric identity, we have: \(\text{cos}(2\theta) = \text{cos}^{2}(\theta) - \text{sin}^{2}(\theta)\).
05
Convert \(\text{cos}(\theta)\) and \(\text{sin}(\theta)\) into rectangular coordinates
Using polar to rectangular transformations: \(\text{cos}(\theta) = \frac{x}{r}\) and \(\text{sin}(\theta) = \frac{y}{r}\). Therefore: \[\text{cos}(2\theta) = \left(\frac{x}{r}\right)^{2} - \left(\frac{y}{r}\right)^{2}\]
06
Simplify \( \text{cos}(2\theta)\) expression
Substitute \(r^{2} = x^{2} + y^{2}\): \(\text{cos}(2\theta) = \frac{x^{2}}{x^{2} + y^{2}} - \frac{y^{2}}{x^{2} + y^{2}}\) Simplify: \[\text{cos}(2\theta) = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}\]
07
Substitute back in the equation
Substitute back to the equation \(x^{2} + y^{2} = \text{cos}(2\theta)\): \[x^{2} + y^{2} = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}\] Multiply both sides by \(x^{2} + y^{2}\): \[ (x^{2} + y^{2})^{2} = x^{2} - y^{2} \]
08
Final equation in rectangular coordinates
Rewriting the equation, we have: \[(x^{2} + y^{2})^{2} = x^{2} - y^{2}\]
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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, are a pair of values \((x, y)\) that determine the location of a point in a plane. These points are plotted using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in the plane can be uniquely identified by its x and y values. In our exercise, we start by converting polar coordinates into rectangular coordinates. We use transformations such as \(x = r \text{cos}(\theta)\) for the x-coordinate and \(y = r \text{sin}(\theta)\) for the y-coordinate to switch between these two systems. This conversion process is essential for simplifying and solving problems like the given equation.
Polar Coordinates
Polar coordinates represent points in a plane using a radius and angle, denoted as \(r\) and \(\theta\) respectively. In this system, \(r\) indicates the distance from the origin to the point, and \(\theta\) is the angle measured counterclockwise from the positive x-axis. For the given equation, \(r^2 = \text{cos}(2\theta)\), we convert the entire polar equation to its rectangular form. By recalling the fundamental transformations between polar and rectangular coordinates, we simplify our task. Particularly, we use \(r^2 = x^2 + y^2\) and later apply it to our equation.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities help simplify calculations in trigonometry. In our problem, the identity \(\text{cos}(2\theta) = \text{cos}^{2}(\theta) - \text{sin}^{2}(\theta)\) is very useful. We recognize \(\text{cos}(2\theta)\) in our equation and use this identity to express it in terms of \( \text{cos}(\theta)\) and \(\text{sin}(\theta)\). By applying transformations like \( \text{cos}(\theta) = \frac{x}{r}\) and \(\text{sin}(\theta) = \frac{y}{r}\), we further translate these trigonometric functions into rectangular coordinates.
Algebraic Manipulation
Algebraic manipulation involves using basic algebra operations to simplify or solve expressions and equations. Once we've converted trigonometric parts of our equation to rectangular coordinates, we need to manipulate the expressions to achieve a solution. Steps include:
- Substituting known values \( r^2 = x^2 + y^2 \) into the equation.
- Simplifying complex fractions, such as \( \frac{x^2}{x^2 + y^2} - \frac{y^2}{x^2 + y^2} \).
- Rewriting equations to isolate the desired variable or expression.