Problem 44
Question
Use a graphing utility to graph the equation. Then answer the given question. $$ \begin{aligned} &r=\frac{3}{2+6 \cos \left(\theta+\frac{\pi}{3}\right)} ; \text { How does the graph differ from the }\\\ &\text { graph of } r=\frac{3}{2+6 \cos \theta} ? \end{aligned} $$
Step-by-Step Solution
Verified Answer
The original graph \( r = \frac{3}{2+6 \cos \left(θ + \frac{\pi}{3} \right)} \) is the same shape as the second one \( r = \frac{3}{2+6 \cos θ}\), but shifted to the left by \( \frac{π}{3} \).
1Step 1: Graph the given polar equation
Firstly, graph the polar equation \( r = \frac{3}{2+6 \cos \left(θ + \frac{\pi}{3} \right)} \). A polar equation describes a curve on the polar coordinate system where an angle and distance from the origin define a point.
2Step 2: Understand each Component of the Equation
In the given equation, \( r \) is the radial distance from the origin (radius) and \( θ \) is the angle, measured counter-clockwise from the positive x-axis. The term \( cos (θ + \frac{π}{3}) \) indicates that graph is shifted by \( \frac{π}{3} \) in the negative direction.
3Step 3: Graph the comparison polar equation
Now, graph the polar equation \( r = \frac{3}{2+6 \cos θ} \). This equation lacks the horizontal shift present in the first equation. Therefore, the graph will be the same shape, but not shifted.
4Step 4: Compare the two Graphs
By comparing both graphs, it can be observed that there is a horizontal shift of \( \frac{π}{3} \) to the left in the original equation compared to the latter equation. Other than this shift, both have the same shape, indicating that addition of \( \frac{π}{3} \) to \( θ \) causes a shift and doesn't affect the shape.
Key Concepts
Polar Coordinate SystemPhase ShiftCosine Function
Polar Coordinate System
The polar coordinate system is an alternative to the rectangular Cartesian coordinate system for representing points on a two-dimensional plane. Unlike the Cartesian system, which uses horizontal (x) and vertical (y) coordinates to define a location, the polar system uses an angle and a distance from a central point, known as the pole (equivalent to the origin in Cartesian coordinates).
Each point in the polar coordinate system is represented by a pair \( (r, \theta) \), where \( r \) is the radial distance from the pole and \( \theta \) is the angular coordinate, typically measured in radians. The angular coordinate defines the direction from the pole in which you'll find the point by moving through an angle relative to the positive x-axis, while \( r \) defines how far along that direction the point lies.
An advantage of representing graphs in polar coordinates is the facility to describe complex curves that are difficult to express in the Cartesian coordinate system. For many types of waves, spirals, and circles, polar equations provide a much more straightforward description. Graphing polar equations is therefore a crucial skill in fields like physics, engineering, and mathematics.
Each point in the polar coordinate system is represented by a pair \( (r, \theta) \), where \( r \) is the radial distance from the pole and \( \theta \) is the angular coordinate, typically measured in radians. The angular coordinate defines the direction from the pole in which you'll find the point by moving through an angle relative to the positive x-axis, while \( r \) defines how far along that direction the point lies.
An advantage of representing graphs in polar coordinates is the facility to describe complex curves that are difficult to express in the Cartesian coordinate system. For many types of waves, spirals, and circles, polar equations provide a much more straightforward description. Graphing polar equations is therefore a crucial skill in fields like physics, engineering, and mathematics.
Phase Shift
A phase shift in mathematics, and more specifically in trigonometry, refers to a horizontal shift in the graph of a trigonometric function. When graphing the function, a phase shift moves the graph to the left or right along the \(\theta\)-axis.
In the polar equation \(r = \frac{3}{2+6 \cos(\theta + \frac{\pi}{3})}\), there is a phase shift of \(-\frac{\pi}{3}\), meaning that the graph of the function is shifted \(\frac{\pi}{3}\) units to the left. This is due to the addition of \(\frac{\pi}{3}\) inside the cosine function's argument. The negative sign indicates the direction of the shift; positive inside the cosine function shifts to the left, and negative would shift to the right.
Understanding phase shifts is essential when comparing two functions that may appear different but are actually the same shape, only shifted horizontally. Phase shifts are not limited to polar graphs and apply to all periodic functions, including sound waves, light waves, and alternating currents.
In the polar equation \(r = \frac{3}{2+6 \cos(\theta + \frac{\pi}{3})}\), there is a phase shift of \(-\frac{\pi}{3}\), meaning that the graph of the function is shifted \(\frac{\pi}{3}\) units to the left. This is due to the addition of \(\frac{\pi}{3}\) inside the cosine function's argument. The negative sign indicates the direction of the shift; positive inside the cosine function shifts to the left, and negative would shift to the right.
Understanding phase shifts is essential when comparing two functions that may appear different but are actually the same shape, only shifted horizontally. Phase shifts are not limited to polar graphs and apply to all periodic functions, including sound waves, light waves, and alternating currents.
Cosine Function
The cosine function is one of the primary trigonometric functions used to describe the relationship between the sides of a right triangle and the angles within it. In a unit circle representation, the cosine of an angle \(\theta\) corresponds to the x-coordinate of a point on the circle's circumference.
The function is periodic, meaning it repeats its values in regular intervals, with a period of \(2\pi\) radians (360 degrees). The standard form of the cosine function is written as \(\cos(\theta)\), where \(\theta\) is the angle. When you modify this angle by adding or subtracting a value, like in \(\cos(\theta + \frac{\pi}{3})\), it affects the starting point of the cosine wave—a concept we link to as phase shift, as previously mentioned.
The graph of a cosine function is a smooth wave that peaks at 1 and troughs at -1, with a regular interval between peaks. It's an even function, which means it is symmetric about the y-axis, and can be used to model many types of periodic phenomena, such as sound waves, electromagnetic waves, and even to describe the shape of certain polar equations, as seen in our exercise.
The function is periodic, meaning it repeats its values in regular intervals, with a period of \(2\pi\) radians (360 degrees). The standard form of the cosine function is written as \(\cos(\theta)\), where \(\theta\) is the angle. When you modify this angle by adding or subtracting a value, like in \(\cos(\theta + \frac{\pi}{3})\), it affects the starting point of the cosine wave—a concept we link to as phase shift, as previously mentioned.
The graph of a cosine function is a smooth wave that peaks at 1 and troughs at -1, with a regular interval between peaks. It's an even function, which means it is symmetric about the y-axis, and can be used to model many types of periodic phenomena, such as sound waves, electromagnetic waves, and even to describe the shape of certain polar equations, as seen in our exercise.
Other exercises in this chapter
Problem 43
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convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes.
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In Exercises \(37-50,\) graph each ellipse and give the location of its foci. $$ \frac{(x-4)^{2}}{4}+\frac{y^{2}}{25}=1 $$
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Explain how to identify the graph of $$A x^{2}+B x y+C y^{2}+D x+E y+F=0$$
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