Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. $$r=2 \sin 5 \theta$$

Short answer

Answer: Desmos or GeoGebra can be used as graphing utilities to check the accuracy of the graph of the polar equation $$r=2\sin 5\theta$$.

Step-by-step solution

Recognize the polar equation

Given the polar equation: $$r=2 \sin 5\theta$$.

Create a table of values

To graph the polar equation, start by creating a table of values for $$\theta$$ and calculate the corresponding $$r$$ values using the given equation. A sample table is shown below: $$\begin{array}{c|c} \theta & r \\ \hline 0 & 0 \\ \pi/10 & 2 \\ 2\pi/10 & 0 \\ 3\pi/10 & -2 \\ 4\pi/10 & 0 \end{array}$$ You can add more points if necessary, but these points should provide enough information to sketch the graph.

Plot the points and sketch the graph

Plot the points from the table on polar coordinate axes. These points will give a general shape of the graph. Sketch the graph based on these points, ensuring that you include any additional details, such as loops, oscillations, or other characteristics.

Use a graphing utility to check work

To ensure accuracy, use a graphing utility (such as Desmos or GeoGebra) to graph the polar equation $$r=2\sin 5\theta$$. Compare this graph to the sketched graph to check for correctness, making any necessary adjustments to the sketch.

Produce a final graph

After checking the work with the graphing utility and making any necessary adjustments, finalize the graph of the polar equation $$r=2\sin 5\theta$$ by ensuring it is clear, accurate, and well-labeled with key points or characteristics.

Key concepts

Graphing Polar Coordinates

Polar coordinates provide a way of graphing that differs from the standard Cartesian system. Instead of using \(x\) and \(y\) values to plot points, polar coordinates use a radius \(r\) and angle \(\theta\). This means each point on the graph is identified by how far it is from the origin and the angle it makes with the positive x-axis.
The polar equation \(r=2\sin 5\theta\) is a great example to learn how plotting in this system works.

• First, identify the angle \(\theta\) and substitute it into the equation to find \(r\).
• Create a table of values for different angles to see the resulting radius, like the sample table provided in the solution. These values tell us the position of each point on the polar plane.
• Start plotting these points on the polar coordinate system, which resembles a series of concentric circles centered at the origin with lines radiating out for angles.

Visualizing the pattern in polar graphs often reveals symmetrical and cyclical properties due to the nature of trigonometric functions.

Trigonometric Functions

Trigonometric functions like sine and cosine play a crucial role in forming polar graphs, as they define how \(r\), the radial distance, changes with \(\theta\), the angle. In the equation \(r = 2\sin 5\theta\),

• The sine function determines the shape and oscillation of the graph. Multiplying \(\theta\) by 5 increases the frequency of oscillation within the interval \(0\) to \(2\pi\).
• The multiple 2 in front of the sine function influences the amplitude, which impacts the maximum distance from the origin reached by the graph.

The sinusoidal nature of \(\sin 5\theta\) results in alternating positive and negative values of \(r\), which creates loops in the graph at different angles. Each loop corresponds to different parts of the function’s positive and negative cycles. This behavior reflects the core property of trigonometric functions—periodicity.

Graphing Utilities

Graphing utilities are incredibly helpful when dealing with complex polar equations. Tools like Desmos and GeoGebra can quickly plot these graphs, confirming what we expect to see from our hand-drawn sketches.

• First, input the equation \(r=2\sin 5\theta\) into the graphing utility.
• Observe the symmetry and the number of loops generated, comparing them to your hand-drawn graph.
• These utilities allow for adjustments, where tweaking parameters or analyzing specific angles becomes simple. This helps in refining our understanding of the graph details.

Using these utilities, students can experiment with different values and instantly see changes, providing immediate feedback and deeper understanding. The ability to visualize mathematical concepts through technology enriches the learning experience and reinforces comprehension.