Question

Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) is a real number and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. \(r=2 \sin 4 \theta\)

Short answer

Question: Graph the polar equation \(r = 2\sin(4\theta)\) and describe its characteristics. Answer: The polar equation \(r = 2\sin(4\theta)\) represents a rose-shaped graph with 4 petals. The petals are located in angle intervals where the radius is greater than zero, which are \(\{(2n)\frac{\pi}{4} < \theta < (2n+1)\frac{\pi}{4}\},\text{ }n = 0, 1, 2, ...\). The graph intersects the polar axis at angles \(\theta = \frac{n\pi}{4}\) for \(n = 0, 1, 2, ...\).

Step-by-step solution

Identify the number of petals

The number of petals in the graph of \(r = a\sin(m\theta)\) is equal to the value of \(m\). Here, we have \(m = 4\). So, the graph will have \(4\) petals.

Calculate the angle interval

To find the angle intervals between the petals, we will divide the full \(360^\circ\) (or \(2\pi\) radians) by the number of petals, which is \(4\). Therefore, the angle interval between petals is \(\frac{2\pi}{4} = \frac{\pi}{2}\) radians (or \(90^\circ\)).

Calculate the points of intersection with the polar axis

We need to find the points where the graph intersects the polar axis. These are the points where \(r = 0\). In order to find these points, we set \(r = 0\) and solve for \(\theta\): $$0 = 2\sin(4\theta)$$ $$\sin(4\theta) = 0$$ The \(\sin\) function is zero at its integer multiples of \(\pi\): $$4\theta = n\pi$$ $$\theta = \frac{n\pi}{4}$$ Where \(n\) is an integer. So, the graph intersects the polar axis at the angles \(\theta = \frac{n\pi}{4}\) for \(n = 0, 1, 2, ...\).

Calculate the angle intervals where the radius is greater than zero

The graph will have a radius greater than zero in the angle intervals where the function \(\sin(4\theta)\) is positive. We can find these intervals by solving for \(\theta\) when \(\sin(4\theta) > 0\). Since \(\sin\) function is positive for \(0 < 4\theta < \pi\), we have: $$0 < \theta < \frac{\pi}{4}$$ And for each additional interval of \(\frac{n\pi}{4}\), \(\sin(4\theta)\) will change the sign. Hence, the angle intervals where the radius is greater than zero are: $$\{(2n)\frac{\pi}{4} < \theta < (2n+1)\frac{\pi}{4}\},\text{ }n = 0, 1, 2, ...$$

Sketch the graph

Now we can sketch the graph using the information we gathered from the previous steps. Draw the polar coordinate system and plot the petals one by one in the angle intervals we found above. Each petal starts at the polar axis intersection point and extends into the widening angle interval where the radius is greater than zero. Make sure to draw a total of \(4\) petals, with an angle interval of \(\frac{\pi}{2}\) radians (or \(90^\circ\)) between them. Your graph should resemble a 4-petaled flower or a rose shape.

Key concepts

Rose Curves

Rose curves are fascinating geometric shapes often encountered in polar coordinates. These curves resemble a flower, hence the name "rose curves." They are created by equations of the form \(r = a\sin(m\theta)\) or \(r = a\cos(m\theta)\).
A couple of things are worth noting about rose curves:

• The parameter \(a\) determines the length of the petals. If \(a\) changes, the size of the petals changes accordingly.
• The parameter \(m\) affects the number of petals. Specifically, if \(m\) is odd, the number of petals is exactly \(m\). If \(m\) is even, the number of petals is \(2m\).

To picture a rose curve, let's consider the example of \(r = 2\sin(4\theta)\). Here, \(m = 4\), but due to the even number, the rose will technically have \(2m = 8\) petals. However, because sine-based equations create patterns that may overlap, we often recognize the visual appearance as a shape with 4 distinct sections. This nuance is part of what makes rose curves both challenging and rewarding to graph.

Graphing Polar Equations

Graphing polar equations such as rose curves involves a few systematic steps. Polar coordinates represent points on a plane using a combination of angles and distances, unlike Cartesian coordinates, which rely on x and y axes.
To graph a rose curve like \(r = 2\sin(4\theta)\), follow these steps:

• Determine the Petals: For our example, since \(m = 4\), expect \(4\) visible petals due to the properties of the sine function.
• Calculate Angle Intervals: Calculate the angle intervals between the petals using the formula \(\frac{2\pi}{m}\). For \(m = 4\), each petal is spaced \(\frac{\pi}{2}\) radians apart.
• Find Intersections: Identify where \(r = 0\) by solving \(\sin(4\theta) = 0\). The solutions show where the polar graph intersects the axis.
• Identify Non-zero Radii Intervals: Determine intervals where \(r > 0\) by solving \(\sin(4\theta) > 0\).

These steps help outline the structure of the curve, making it easier to sketch accurately.

Trigonometric Functions

Trigonometric functions such as sine and cosine are vital in graphing polar equations like rose curves. Let's consider why these functions are so important in polar graphs.

• The amplitude, determined by the multiplicative factor (e.g., the '2' in \(2\sin(4\theta)\)), affects the radius, which is the distance from the pole in polar coordinates.
• The function's behavior, like where sine is zero or positive, defines key intervals for graphing. For instance, \(\sin(x) = 0\) at integer multiples of \(\pi\), which assists in figuring out intersections and shape of the rose curves.
• Trigonometric functions also introduce periodicity, repeating their values at regular intervals. This helps in predicting pattern repetitions in polar graphs.

Understanding how trigonometric functions operate within polar equations is crucial to successfully sketching curves like the rose, revealing beautiful periodic and symmetrical designs.