Problem 19 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=2 \cos 4 \theta\) (eight- leafed rose)

Step-by-Step Solution

Answer

The graph of \( r = 2 \cos 4\theta \) is an eight-leafed rose, with petals symmetrically placed around the origin at \( \frac{\pi}{4} \) radians apart.

Step 1: Identify the type of polar equation

The given equation is in the form of a rose curve, which can be identified because it has the structure of \( r = a \, \cos(k\theta) \) where \( k \) is even.

Step 2: Determine the number of petals

For the cosine rose curve \( r = a \cos(k\theta) \), the number of petals is given by \(2k\) when \( k \) is even.Here, \( k = 4 \), so the number of petals is \( 2 \times 4 = 8 \).

Step 3: Sketch one petal

To start, sketch one petal of the rose curve. For \(\theta = 0\), \( r = 2 \cos(4 \times 0) = 2 \). This point is at (2, 0). The petals will be symmetrically placed around the origin.

Step 4: Plot remaining petals

Use symmetry to plot the remaining petals around the origin. Since there are 8 petals, each petal will be spaced \( \frac{2\pi}{8} = \frac{\pi}{4} \) radians apart.

Step 5: Complete the sketch

Once all 8 petals are plotted at every \(\frac{\pi}{4}\) interval, connect the petals to form the complete eight-leafed rose curve.

Key Concepts

rose curve

Understanding a rose curve in polar coordinates helps in visualizing fascinating and symmetric shapes.
Rose curves are named for their petal-like structures. They follow a specific polar equation form: \( r = a \, \cos(k\theta) \) or \( r = a \, \sin(k\theta) \).
Here, 'a' determines the length of each petal, and 'k' affects the number and symmetry of the petals.
When 'k' is even, the curve generates \( 2k \) petals. For odd values of 'k', only 'k' petals form.
The given problem entails drawing \( r = 2 \, \cos(4\theta) \) implying an eight-leafed rose as \( k = 4 \). Such curves illustrate how simple trig functions can create intricate patterns.

• Tip: Pay close attention to 'k' and whether it is even or odd to anticipate the number of petals correctly.

cosine function

The cosine function is crucial in various mathematical concepts, including polar coordinates.
In this problem, \(2 \cos(4\theta)\) showcases the utility of cosine in shaping complex spirals.
Cosine, abbreviated as 'cos,' is a trigonometric function representing the adjacent side's ratio to the hypotenuse of a right triangle.
Moving to polar equations, substituting 'cos' impacts the graph's symmetry.
The function \( \cos(k\theta) \) oscillates between -1 and 1, stretching the radius 'r' accordingly.

• Note: Positive values of the cosine function orient petals directly, while negative values reflect over the pole.
• Example: At \( \theta = 0 \), \( \cos(4 \times 0) = 1 \) gives the point (2,0), indicating the petal's starting position.

polar equation

Polar equations describe curves using a radius and an angle.
Unlike Cartesian coordinates, polar coordinates involve 'r' (radius) and 'θ' (angle), providing a unique way to draw curves.
For \( r = 2 \cos(4\theta) \), the equation's two components offer insights:

• Radius (r): Distance from the origin.

• Angle (θ): Measurement from the positive x-axis.

This form's beauty lies in its ability to create patterns not achievable with Cartesian equations. By plotting points for varied 'θ', you build the rose curve step-by-step, showing intricate petals.

• Reminder: Practice converting between Cartesian and polar forms to improve understanding and skills.