Question

Use technology to plot \(r=\sin \left(\frac{3 \theta}{7}\right)\) (use the interval \(0 \leq \theta \leq 14 \pi\) ).

Short answer

Use a polar graphing tool to plot \( r = \sin \left( \frac{3\theta}{7} \right) \) for \( 0 \leq \theta \leq 14\pi \). Observe the repeating loop pattern.

Step-by-step solution

Understanding the Polar Equation

The function given is \( r = \sin \left( \frac{3\theta}{7} \right) \). This is a polar equation where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle. The function defines how \( r \) changes as \( \theta \) changes, tracing a curve in the polar coordinate plane.

Setting Up the Plotting Tool

To plot this equation, select software that can handle polar plots such as Desmos, Geogebra, or a graphing calculator (like TI-89). These tools have options to plot functions in polar form.

Inputting the Equation

Enter the equation \( r = \sin \left( \frac{3\theta}{7} \right) \) into the plotter. Ensure you are in polar mode by selecting the appropriate setting in your tool.

Adjusting the Theta Range

Set the range for \( \theta \) from \( 0 \) to \( 14\pi \). This can typically be done by specifying the domain for \( \theta \) in your software. This range ensures that you capture the full behavior of the function as specified.

Analyzing the Plot

Once the plot is generated, observe the pattern of the curve. Given the function's form, it will likely have a symmetrical or repeating pattern. Look for how the \( \sin \) function impacts the shape, causing it to form loops around the origin.

Key concepts

Polar Equation Plotting

Plotting polar equations is an exciting mathematical task. A polar equation like \( r = \sin \left( \frac{3\theta}{7} \right) \) expresses a relationship between the radius \( r \) and angle \( \theta \) in a circular fashion, unlike rectangular coordinates that rely on \( x \) and \( y \). Instead of focusing on horizontal and vertical distances, polar equations concern themselves with angles and distances from a central point, known as the pole.
Polar coordinates are particularly useful in scenarios where natural circular symmetry is present. To effectively plot a polar equation, it is important to thoroughly understand how changes in \( \theta \) affect the radius \( r \), and consequently the shape of the traced path or curve.
For this exercise, as \( \theta \) swings through the specified interval, \( 0 \leq \theta \leq 14\pi \), observe how \( r \) varies. The polar plot captures curves where the radius lengthens and shortens, forming fascinating loops and symmetrical patterns around the pole.

Trigonometric Functions

Trigonometric functions are fundamental to understanding polar equations. The given exercise uses the sine function, which is one of the six basic trigonometric functions used in mathematics to express repetitive or wave-like phenomena. In the context of polar coordinates, the sine function influences the radial distance \( r \) based on the angle \( \theta \).
When we analyze \( r = \sin \left( \frac{3\theta}{7} \right) \), it's the sine function dictating how \( r \) behaves. The argument of the sine function, \( \frac{3\theta}{7} \), introduces a transformation that affects the frequency and phase shift of the pattern. In simple terms, this breakdown explains why the curve might loop multiple times as \( \theta \) progresses through its range.
Understanding how trigonometric functions like sine, cosine, and tangent transform under various manipulations is key to mastering polar coordinates. These functions enable us to visualize and predict how the polar graph will look, offering insights into its symmetry and periodicity.

Graphing Technology

To graph the polar equation \( r = \sin \left( \frac{3\theta}{7} \right) \), leveraging graphing technology simplifies the process remarkably. Tools like Desmos, Geogebra, and advanced graphing calculators such as the TI-89 are perfect for this task.
These graphing tools are equipped with capabilities to switch between different coordinate systems, allowing users to plot equations directly in polar form. It is essential to ensure that the software or calculator is in polar mode when inputting the equation. Different tools might require slightly different operations for setting up the polar plot, but typically, selecting the polar grid or entering settings to define \( \theta \)'s range is straightforward.
Follow the steps: Enter the equation in the proper field, define the range of \( \theta \) to be from \( 0 \) to \( 14\pi \), and then allow the technology to generate the graph. Analyzing the resulting plot can reveal complex and beautiful patterns, providing deep visual understanding alongside theoretical knowledge.