Problem 40 - Precalculus

Question

Graph equation. \(r=5 \cos (2 \theta)\)

Step-by-Step Solution

Answer

The graph represents a rose curve with 4 petals, each corresponding to \(r=5\) in polar coordinates.

Step 1: Understanding the Polar Equation

The equation given is a polar equation of the form \(r = 5 \cos(2\theta)\). In polar coordinates, \(r\) represents the radius, or distance from the origin, and \(\theta\) represents the angle from the positive x-axis. The term \(2\theta\) indicates that the graph will have rotational symmetry.

Step 2: Identify the Type of Graph

The equation is of the form \(r = a \cos(b\theta)\), which is a standard form for a rose curve. A rose curve of the form \(r = a \cos(b\theta)\) with \(b = 2\) will have \(2b = 4\) petals if \(b\) is even. Thus, the graph will have 4 petals.

Step 3: Determine the Key Values of \(\theta\)

To plot the graph, identify the key values of \(\theta\) where \(r\) will be 0, maximum, or minimum. Since \(r = 5 \cos(2\theta)\), the graph will have maximum radius of 5 when \(\cos(2\theta) = 1\) and a minimum of 0 when \(\cos(2\theta) = 0\). This occurs when \(2\theta = 0, \pi, 2\pi,\) etc.

Step 4: Translating Key Values into Plots

Using the symmetry and periodic nature of the cosine function, calculate \(r\) for specific values of \(\theta\):- At \(\theta = 0\), \(r = 5 \cos(0) = 5\)- At \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\), \(r = 5 \cos\left(\frac{\pi}{2}\right) = 0\)- Repeat for \(\theta = \pi\), \(r = 5 \cos(2\pi) = 5\) and so on, to complete all petals.

Step 5: Plotting the Rose Curve

Plot the points calculated in Step 4 on polar graph paper and connect them smoothly to form the rose. Since the function \(\cos\) is periodic, the petals will repeat every \(\pi\), and there will be 4 symmetrical petals in total, confirming the rose curve pattern.

Key Concepts

Rose Curve

The rose curve is a fascinating type of graph in polar coordinates. It gets its name due to the petal-like shapes it forms. These curves are defined by equations of the form \( r = a \cos(b\theta) \) or \( r = a \sin(b\theta) \). Here, \( a \) controls the length of the petals, and \( b \) determines the number of petals.

• If \( b \) is odd, the curve will have exactly \( b \) petals.
• If \( b \) is even, the number of petals is \( 2b \).

This property leads to interesting and symmetric designs when plotted. In the given exercise, since \( b = 2 \), there will be five petals.Understanding rose curves helps in grasping the beauty of mathematical symmetry and periodicity. Each petal reflects the underlying trigonometric function's behavior as it cycles through its range.

Polar Coordinates

Polar coordinates are a powerful way to describe the position of a point in the plane using two values: a radius \( r \) and an angle \( \theta \). Unlike Cartesian coordinates, which use \( x \) and \( y \) values for positioning, polar coordinates locate a point based on:

• The distance \( r \) from the origin, or pole.
• The angle \( \theta \) measured from the positive x-axis.

This system is particularly useful for graphing equations that have circular or spiral structures. It's essential to understand that each point in polar coordinates can have multiple representations, due to the periodic nature of the angle \( \theta \). For example, \( (r, \theta) \) is the same as \( (r, \theta + 2\pi) \).Polar coordinates simplify the process of plotting curves like circles, spirals, and the rose curve by converting the complexities of these shapes into straightforward equations.

Graphing in Polar Form

Graphing in polar form transforms the way we visualize mathematical equations. When plotting a polar equation, we use the radius \( r \) as a function of the angle \( \theta \). Unlike graphing in the Cartesian plane, you are looking at the dynamics between angle and distance.To graph in polar form, you:

• Identify key angles where the function reaches its maximum, minimum, or crosses the pole \( r = 0 \).
• Calculate the corresponding \( r \) values for those angles.
• Draw each point on a polar grid using its \( r \) and \( \theta \) values.

For the rose curve with \( r = 5 \cos(2\theta) \), the petals repeat every \( \pi \), making it easy to predict the curve's symmetry and complete the graph. Polar graphs often reveal patterns and symmetries more beautifully and intuitively than their Cartesian counterparts.Mastering graphing in polar form can enhance your skills in analyzing complex patterns, particularly those involving oscillations and periodic functions.