Problem 49 - Algebra and Trigonometry

Question

47–50 Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$\left(x^{2}+y^{2}\right)^{2}=x^{2}-y^{2}$$

Step-by-Step Solution

Answer

The polar graph of the equation resembles a rose with four petals.

Step 1: Convert Rectangular to Polar Coordinates

In polar coordinates, we know that \( x = r\cos\theta \) and \( y = r\sin\theta \). Substituting these into the equation \( (x^{2}+y^{2})^{2}=x^{2}-y^{2} \), we get:\( (r^{2})^{2} = (r\cos\theta)^{2} - (r\sin\theta)^{2} \).Simplify to get:\( r^{4} = r^{2}\cos^{2}\theta - r^{2}\sin^{2}\theta \) which simplifies to \( r^{4} = r^{2} (\cos^{2}\theta - \sin^{2}\theta) \).

Step 2: Simplify the Polar Equation

Factor and simplify the polar equation:Divide both sides by \( r^{2} \) (assuming \( r eq 0 \)):\( r^{2} = \cos^{2}\theta - \sin^{2}\theta \).Recall that \( \cos^{2}\theta - \sin^{2}\theta = \cos(2\theta) \). So we have:\( r^{2} = \cos(2\theta) \).Thus, \( r = \pm \sqrt{\cos(2\theta)} \).

Step 3: Analyze the Polar Equation

In the polar form \( r = \pm \sqrt{\cos(2\theta)} \), graphically analyze the equation to understand the shape.The equation stipulates that \( r \) is real only when \( \cos(2\theta) \geq 0 \), which implies that \( 2\theta \) must be in the range where \( \cos \) is non-negative, i.e., in intervals \([0, \frac{\pi}{2}]) \cup [\pi, \frac{3\pi}{2}]\).

Step 4: Sketch the Graph Based on Analysis

Within the valid ranges of \( 2\theta \), the values of \( r \) alternate between positive and negative, forming symmetrical lobes.Reflect the structure of the polar plot in the Cartesian plane corresponding to these specific \( 2\theta \) ranges.The polar graph has symmetry and reflects around the origin with lobes appearing along the axis in the specified ranges due to the periodicity of cosine.

Key Concepts

Rectangular to Polar Conversion

Converting from rectangular to polar coordinates is a useful skill in mathematics, especially when dealing with certain types of graphs or equations that are more easily expressed in polar form. The key relationships to note are:

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)

This means that any point in the rectangular coordinate system \((x, y)\) can be expressed in terms of \((r, \theta)\) in the polar system, where \(r\) is the distance from the origin to the point and \(\theta\) is the angle formed with the positive x-axis.
To convert a rectangular equation into a polar equation, substitute these expressions for \(x\) and \(y\). For example, in our equation \((x^{2} + y^{2})^{2} = x^{2} - y^{2} \), substituting the polar equivalents results in an equation that can be simplified further. This simplification often reveals symmetry or periodic behavior that is not immediately apparent in the rectangular form.
Understanding these conversions is crucial when you need to sketch or analyze graphs that are easier to manage in the polar format.

Graphing Polar Equations

Graphing polar equations is an intriguing exercise that often results in beautiful and complex shapes. Once an equation is simplified in polar form, like \( r = \pm \sqrt{\cos(2\theta)} \), the next step is to graph it to identify its shape and characteristics.
The form \( r = \pm \sqrt{\cos(2\theta)} \) suggests a behavior that depends on the angle \(\theta\), primarily since \(\cos(2\theta)\) dictates when \(r\) is real and manageable. When graphing, you need to account for where the cosine term is non-negative, because radians of \(\theta\) outside this range would yield imaginary or non-real values of \(r\).
Graphing involves plotting points for various values of \(\theta\) within accepted intervals, typically where \(r\) is defined, which are from

• \([0, \frac{\pi}{2}]\)
• \([\pi, \frac{3\pi}{2}]\)

The resulting plot will show symmetric lobes that reflect about the axis and origin, demonstrating the periodic nature of \(\cos(2\theta) \). By understanding the interplay of \(\theta\) and \(r\), you can accurately capture the entire behavior of the polar graph.

Symmetry in Polar Graphs

Symmetry plays a critical role in understanding and simplifying polar graphs. When examining an equation like \( r = \pm \sqrt{\cos(2\theta)} \), identifying symmetry helps predict and confirm the structure of the graph without having to calculate every possible point.
There are typical types of symmetry in polar graphs:

• Symmetry about the polar axis (x-axis in rectangular coordinates).
• Symmetry about the line \(\theta = \frac{\pi}{2}\) (y-axis in rectangular coordinates).
• Symmetry about the origin.

The equation's behavior tells you it has symmetry around the origin and axis, primarily because of the even function \(\cos(2\theta)\). The periodicity of \(\cos(2\theta)\) also contributes, as it repeats its values every \(\pi\) radians. This means the shapes or 'lobes' repeat in the intervals outlined earlier, creating a reflection of
the graph through the origin, such that if a lobe appears in one half of the graph, its symmetrical counterpart will appear in the opposite half. Identifying symmetry early on can significantly reduce the work required in graphing a polar function, enabling more focused calculation of points where necessary.