Question

Consider the polar curve $r=\cos (n\theta /m)$ where $n$ and $m$ are integers. a. Graph the complete curve when $n=2$ and $m=3$ b. Graph the complete curve when $n=3$ and $m=7$ c. Find a general rule in terms of $m$ and $n$ (where $m$ and $n$ have no common factors) for determining the least positive number $P$ such that the complete curve is generated over the interval $[0,P]$

Short answer

Describe the general rule for determining the least positive number P over which the complete curve is generated in the polar equation r = cos(nθ/m). The general rule for determining the least positive number P over which the complete curve is generated in the polar equation r = cos(nθ/m) is given by: P = (2πm) / n This rule ensures that the argument inside the cosine function completes one period, resulting in a complete curve.

Step-by-step solution

Understanding Polar Coordinates

Polar coordinates are a way to describe the location of a point in a plane using the distance from a reference point (the origin) and the angle between a reference direction and the point. A point can be represented as $(r,\theta )$, where $r$ is the radial distance from the origin and $\theta$ is the angle made with the positive x-axis (measured counterclockwise). In our case, we are given the polar equation $r=\cos (\frac{n\theta }{m})$, which describes how the radial distance $r$ changes as the angle $\theta$ varies.

General Methodology for Graphing Polar Curves

To graph the curve defined by the polar equation, we will follow these steps: 1. Identify the values of $n$ and $m$ for the specific curve. 2. Determine the least positive angle $P$ over which the complete curve is generated. 3. Divide the interval $[0,P]$ into a reasonable number of sub-intervals. 4. Calculate the radial distances $r$ for each angle in the sub-intervals using the given polar equation. 5. Plot the points $(r,\theta )$ and join them to create the curve. Now let's find the general rule for determining the least positive number $P$.

General Rule for Determining the Least Positive Number $P$

To find the general rule for determining the least positive number $P$, we need to consider the properties of the cosine function. The cosine function is periodic, with a period of $2\pi$. Therefore, we will have a complete curve when the argument inside the cosine function, $\frac{n\theta }{m}$, completes one period. We have: $\frac{n\theta }{m}=2\pi k$, where $k$ is an integer. Then, solving for $\theta$: $\theta =\frac{2\pi mk}{n}$ Since we are trying to find the least positive number $P$, it is sufficient to consider when $k=1$. So, $P=\frac{2\pi m}{n}$. Now, let's apply the methodology to graph the curve for given values of $n$ and $m$ in parts a and b.

Graph the Complete Curve when $n=2$ and $m=3$

1. The values of $n$ and $m$ are given: $n=2$ and $m=3$. 2. Using the general rule for $P$, we have: $P=\frac{2\pi m}{n}=\frac{2\pi (3)}{2}=3\pi$. 3. Divide the interval $[0,3\pi ]$ into, for example, 12 equal sub-intervals. 4. Calculate the radial distances $r$ for each angle in the sub-intervals using the given polar equation: $r=\cos (\frac{2\theta }{3})$. 5. Plot the points $(r,\theta )$ and join them to create the curve. Now let's graph the complete curve when $n=3$ and $m=7$.

Graph the Complete Curve when $n=3$ and $m=7$

1. The values of $n$ and $m$ are given: $n=3$ and $m=7$. 2. Using the general rule for $P$, we have: $P=\frac{2\pi m}{n}=\frac{2\pi (7)}{3}=\frac{14\pi }{3}$. 3. Divide the interval $[0,\frac{14\pi }{3}]$ into, for example, 14 equal sub-intervals. 4. Calculate the radial distances $r$ for each angle in the sub-intervals using the given polar equation: $r=\cos (\frac{3\theta }{7})$. 5. Plot the points $(r,\theta )$ and join them to create the curve. In conclusion, we have graphed the polar curves for the given values of $n$ and $m$ by following a general methodology and derived a rule for determining the least positive number $P$ over which the complete curve is generated.

Key concepts

Graphing Polar Curves

Graphing polar curves can be a rewarding way to explore mathematics visually. In polar coordinates, each point is defined by a distance from the origin (radial distance) and an angle. When graphing polar curves, such as the equation $r=\cos (\frac{n\theta }{m})$, you are essentially plotting how the distance changes with each angle. This process allows us to visualize the beautiful symmetry and periodic behavior of polar equations.

To graph a polar curve effectively, start by setting values for $n$ and $m$. Then, follow these steps:

• Determine the total angle needed to sketch the full curve, which we refer to as $P$.
• Divide the angle range from $[0,P]$ into smaller segments. This gives you a set of angles to substitute into the function.
• For each angle, calculate the corresponding $r$ using the equation $r=\cos (\frac{n\theta }{m})$.
• Plot the points $(r,\theta )$ on the polar coordinate system and connect these points to show the curve.

By systematically following these steps, the polar curve is effectively graphed, exhibiting patterns and symmetries unique to chosen values of $n$ and $m$.

Periodicity of Cosine Function

One of the fascinating aspects of polar curves comes from the periodic nature of the cosine function itself. The cosine function, $\cos (\theta )$, is periodic, repeating its values at regular intervals of $2\pi$. This quality translates to polar coordinates as well, where the function's period affects the length of the curve before it starts to repeat.

In the context of our polar equation $r=\cos (\frac{n\theta }{m})$, the expression $\frac{n\theta }{m}$ inside the cosine function influences the period of the graph. To find when the cosine function completes a full cycle, we set $\frac{n\theta }{m}=2\pi k$. Solving for $\theta$ gives us:

$$\theta =\frac{2\pi mk}{n}$$

For the graph to complete one cycle and display all unique points, setting $k=1$ provides the least positive interval, which we denote as $P$, calculated as $\frac{2\pi m}{n}$. Hence, the periodic behavior of cosine directly influences when a polar curve starts repeating its pattern.

Polar Equations

Polar equations describe relationships between the radius $r$ and the angle $\theta$. They offer a distinctive way to represent and visualize functions, diverging from the traditional Cartesian coordinate system. The equation $r=\cos (\frac{n\theta }{m})$ is one such polar equation, where the radius is expressed as a function of the angle.

These equations allow us to graph complex curves by varying just two parameters: $n$ and $m$. By experimenting with different integers for $n$ and $m$, polar equations provide insights into fascinating geometric patterns and symmetries.

• Increasing the value of $n$ generally results in more loops or petals in the graph, as it increases the angular frequency.
• Changing $m$ usually alters the curve's width and spacing.

Using polar equations involves exploring these parameters to uncover curves with desirable characteristics. Each plot of a polar equation possesses intrinsic beauty, generated by the mathematical harmony between radial distances and angles.