Problem 61
Question
Use a graphing utility to graph \(r_{1}=2 \sin (3 \theta)\) and \(r_{2}=2 \sin \left(3\left(\theta+\frac{\pi}{3}\right)\right) .\) Explain the relationship between the two graphs in terms of rotations.
Step-by-Step Solution
Verified Answer
The graph \(r_{2}=2 \sin \left(3\left(\theta+\frac{\pi}{3}\right)\right)\) is a rotation of the graph \(r_{1}=2 \sin (3 \theta)\). The rotation is anticlockwise by \( \frac{\pi}{3} \) radians.
1Step 1: Graph the first polar function
Graph the first polar function \(r_{1}=2 \sin (3 \theta)\). Note the pattern and shape of the plot. Typical software for this task are graphic calculator, Desmos, or Geogebra. Points will be plotted in a form of a rose with 3 petals.
2Step 2: Graph the second polar function
Graph the second polar function \(r_{2}=2 \sin \left(3\left(\theta+\frac{\pi}{3}\right)\right)\). As \( \theta \) shifts by \( \frac{\pi}{3} \), it will cause a rotation. And observe the plot to find the difference.
3Step 3: Compare the two graphs and explain the relationship
Compare the two plots from Step 1 and Step 2. The difference between the two is that the second graph is rotated anti-clockwise by \( \frac{\pi}{3} \) radians compared to the first one. Hence the \( \frac{\pi}{3} \) inside the sine function of the second equation is responsible for this rotation.
Key Concepts
Polar FunctionsGraphing UtilityTrigonometric TransformationsGraph Comparison
Polar Functions
Polar functions are mathematical expressions that define curves by specifying distances and angles. Each point on a polar curve is determined by an angle \( \theta \) and a distance \( r \) from the origin, also known as the pole. Unlike Cartesian coordinates, polar coordinates rely on the relationship between these angles and distances to form various shapes, such as circles, spirals, and roses.
In the context of our exercise, we are dealing with rose curves, which are distinguished by their petal-like shapes. The equation \( r = 2 \sin(3\theta) \) is an example of a rose curve where the number of petals is determined by the coefficient of \( \theta \). Here, it's 3, indicating the function will yield a rose with three symmetrical petals.
In the context of our exercise, we are dealing with rose curves, which are distinguished by their petal-like shapes. The equation \( r = 2 \sin(3\theta) \) is an example of a rose curve where the number of petals is determined by the coefficient of \( \theta \). Here, it's 3, indicating the function will yield a rose with three symmetrical petals.
Graphing Utility
A graphing utility is a software tool used to visualize mathematical functions by plotting them on a graph. These tools can range from physical hand-held calculators to sophisticated online platforms like Desmos or Geogebra. They assist in easily plotting complex functions and observing their behavior.
When graphing polar functions like \( r = 2 \sin(3\theta) \), these utilities allow users to input the equation in polar form, setting the theta range appropriately. The output is a visual representation that helps in understanding the geometric properties of the function. This is particularly helpful with polar functions, which often yield intricate and non-intuitive shapes.
When graphing polar functions like \( r = 2 \sin(3\theta) \), these utilities allow users to input the equation in polar form, setting the theta range appropriately. The output is a visual representation that helps in understanding the geometric properties of the function. This is particularly helpful with polar functions, which often yield intricate and non-intuitive shapes.
Trigonometric Transformations
Trigonometric transformations involve changes to the standard trigonometric functions, such as shifts, stretches, or rotations. These transformations alter the position or shape of a graph.
In our exercise, the function \( r_2 = 2 \sin \left(3\left(\theta + \frac{\pi}{3}\right)\right) \) demonstrates a rotation transformation. The addition of \( \frac{\pi}{3} \) within the angle shifts the entire graph, effectively rotating the shape on the polar coordinate system by \( \frac{\pi}{3} \) radians. This concept is crucial for understanding how modifications within the function can shift the positions of the graph's features without altering its fundamental shape.
In our exercise, the function \( r_2 = 2 \sin \left(3\left(\theta + \frac{\pi}{3}\right)\right) \) demonstrates a rotation transformation. The addition of \( \frac{\pi}{3} \) within the angle shifts the entire graph, effectively rotating the shape on the polar coordinate system by \( \frac{\pi}{3} \) radians. This concept is crucial for understanding how modifications within the function can shift the positions of the graph's features without altering its fundamental shape.
Graph Comparison
Graph comparison involves examining two or more graphs to identify their differences and similarities. It's a vital step in understanding the impact of modifications on mathematical functions.
In our exercise, comparing the graphs of \( r_1 = 2 \sin(3\theta) \) and \( r_2 = 2 \sin \left(3\left(\theta + \frac{\pi}{3}\right)\right) \), the key difference is the rotational shift by \( \frac{\pi}{3} \) in the second graph. Despite this rotation, the number of petals and their shape remains consistent. This observation highlights the role of the angular transformation in polar functions.
When comparing graphs, ensure to look out for rotational shifts, reflection, or scaling, as these reveal how parameter changes influence visual outcomes in polar functions.
In our exercise, comparing the graphs of \( r_1 = 2 \sin(3\theta) \) and \( r_2 = 2 \sin \left(3\left(\theta + \frac{\pi}{3}\right)\right) \), the key difference is the rotational shift by \( \frac{\pi}{3} \) in the second graph. Despite this rotation, the number of petals and their shape remains consistent. This observation highlights the role of the angular transformation in polar functions.
When comparing graphs, ensure to look out for rotational shifts, reflection, or scaling, as these reveal how parameter changes influence visual outcomes in polar functions.
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