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Chapter 45: Problem 924

Graph r = sin 3?.

Short Answer

Expert verified
To graph the polar function \(r = \sin(3\theta)\), first note that the period is \(\frac{2\pi}{3}\). Calculate the values of r for a few values of 胃 within the interval \(0 \le \theta \le \frac{2\pi}{3}\) and plot the points on the polar graph. Notice the 3-fold symmetry and maximum/minimum values of r (1 and -1). Connect the points, and extend the graph by making two more copies of the curve and rotating them by 120 and 240 degrees. The final graph will have a 3-petal shape.

Step by step solution

01

Identify key points and intervals for plotting

Begin by analyzing the given function, r = sin(3胃). Since the sine function has a period of 2蟺, the period of sin(3胃) will be 2蟺/3. This means we only need to plot points in the interval 0 鈮� 胃 鈮� 2蟺/3.
02

Tabulate the values of r for some values of 胃

Calculate r = sin(3胃) for a few values of 胃 within the interval 0 鈮� 胃 鈮� 2蟺/3. For this function, we can choose the following values of 胃: 0, 蟺/6, 蟺/3, and 蟺/2 (which corresponds to 0, 30, 60, and 90 degrees, respectively). 胃 | r ---|--- 0 | sin(0) = 0 蟺/6 | sin(蟺/2) = 1 蟺/3 | sin(蟺) = 0 蟺/2 | sin(3蟺/2) = -1
03

Plot the points on a polar graph paper

Using the calculated values of r and 胃 from the table, plot the points on a polar graph paper. Then, notice any symmetry and other notable points in the plotted graph. The graph will have 3-fold symmetry, which means it will have three identical parts within the full circle around the origin. Additionally, the maximum value of r is 1, and the minimum value is -1.
04

Connect the points and extend the graph

Connect the points plotted in the previous step to form a curve. Since the graph has a 3-fold symmetry, you can extend the curve by making two more copies of the first curve and rotating them by 120 and 240 degrees around the origin, respectively. By connecting the points and extending the graph, you should now have the complete graph of r = sin(3胃). The graph should have a 3-petal shape, with one petal in each of the first, second, and third quadrants.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions play an important role in mathematics, especially when dealing with angles and dimensions. Sine, cosine, and tangent are the core trigonometric functions commonly used. In this exercise, we focus on the sine function, represented as \( \sin(x) \). Triangular waveforms, circles, and oscillating motions can be described effectively using sine and other trigonometric functions, often providing a foundation for analyzing periodic phenomena.
  • The sine function yields values ranging from -1 to 1.
  • It is periodic, repeating its pattern over a specific interval.
  • Sine is key to calculating angles in right triangles, being the ratio of the opposite side to the hypotenuse.
Understanding how these functions interact and transform helps us graph and analyze various mathematical problems.
Polar Graphing
Polar graphing is a way to represent mathematical functions and equations on a plane using polar coordinates. Unlike the Cartesian coordinate system, which uses x and y, polar graphing relies on radius \( r \) and angle \( \theta \). To plot a polar graph, you identify points based on their distance from the origin and the angle they make with the positive x-axis.
In this context, we're plotting \( r = \sin(3\theta) \). We calculate \( r \) for different angles \( \theta \) and then plot these values. This creates an image on polar graph paper. One of the characteristics of polar graphs is symmetry, often providing a visually appealing representation of trigonometric functions.
  • Polar graphs can represent circles, spirals, and rose patterns.
  • The origin is the center point where all radii meet.
  • Curves are formed by connecting points derived from the function.
Angular Symmetry
Angular symmetry refers to the balanced and repetitive parts of a graph around an angle or axis. In polar coordinates, angular symmetry can make a graph quite simple to interpret and understand.
When dealing with \( r = \sin(3\theta) \), the graph exhibits rotational symmetry. Specifically, this graph shows three-fold symmetry, meaning it looks the same with a 120-degree rotation. Graphs with such symmetry are called rose curves.
  • Symmetric graphs have repetitive segments or parts.
  • Rotational symmetry allows for duplicating sections across angles.
  • Identifying symmetry helps extend the graph without recalculating each point.
Period of Sine Wave
The period of a function describes how long it takes for the function to complete a full cycle of its pattern. For the sine function \( \sin(x) \), the standard period is \( 2\pi \). However, when sine is modified, such as \( \sin(3\theta) \), the period changes.
In \( r = \sin(3\theta) \), the period is calculated as \( \frac{2\pi}{3} \). This is because the frequency of the waveform has been scaled by 3, compressing its cycle.
  • A wave with a higher frequency will have a shorter period.
  • The period tells us how wide a single cycle of the function spans on the graph.
  • Understanding the period aids in accurately plotting and interpreting graphs.
Recognizing the period helps greatly in graphing and predicting the behavior of trigonometric functions within particular intervals.

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Most popular questions from this chapter

Transform the equation \(\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{x}+3 \mathrm{y}=3\) to a polar equation.

Transform the equation \(\mathrm{p}=2 \cos \theta\) to rectangular coordinates.

Transform the equation r = 4/(2 ? 3 sin ?) to an equation cartesian coordinates.

Find the conditions which the equation of a polar figure must satisfy for the figure to be (a) symmetric about the polar axis, (b) symmetric about the 90掳?axis, (c) symmetric about the pole.

Convert the equation r = tan ? + cot ? to an equation in cartesian coordinates.
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