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Magazine Chapter 13: Problem 22
Sketch the graph of the polar equation. $$ r=4 \cos ^{2} \frac{1}{2} \theta $$
Short Answer
Expert verified
The graph is a polar rose with two petals, symmetric about the polar axis. It stretches from \( r = 4 \) at \( \theta = 0 \) and returns to \( r = 4 \) at \( \theta = 2\pi \).
Step by step solution
01
Understand the Polar Equation
The given polar equation is \( r = 4 \cos^{2} \left(\frac{1}{2} \theta\right) \). This means that the radial distance \( r \) depends on the angle \( \theta \) through the cosine function squared. The \( \cos^{2} \left(\frac{1}{2} \theta\right) \) suggests a trigonometric pattern that will be repeated or transformed as \( \theta \) changes.
02
Identify Symmetry
Notice that the given polar equation \( r = 4 \cos^{2} \left(\frac{1}{2} \theta\right) \) is symmetric about the polar axis (horizontal line through the origin) because squaring cosine makes it non-negative in all quadrants.
03
Determine Key Points
Since \( r \) is a function of \( \cos^{2} \left(\frac{1}{2} \theta\right) \), analyze key angles \( \theta \):- At \( \theta = 0 \), \( r = 4 \cos^{2}(0) = 4 \).- At \( \theta = \pi \), \( r = 4 \cos^{2}\left(\frac{\pi}{2}\right) = 0 \).- At \( \theta = 2\pi \), \( r = 4 \cos^{2}(\pi) = 4 \).These calculations imply the graph passes through \( (4, 0) \) and returns to \( (4, 2\pi) \), completing a cycle.
04
Graph the Polar Equation
Plot the points calculated in Step 3, noting that as \( \theta \) increases from 0 to \( 2\pi \), \( r \) varies from 4 to 0 back to 4. Since the formula involves \( \cos^2 \), the graph describes a petal-like shape.- Begin at \( r = 4 \) when \( \theta = 0 \)- End at \( r = 4 \) when \( \theta = 2\pi \)The graph is a type of polar rose with petals closing at \( \theta = \pi \).
05
Draw the Complete Graph
Based on periodicity and symmetry, sketch the graph by connecting the dots and reflecting the structure. The graph has a petal shape due to \( \cos^2 \), consisting of two symmetrical roses (petals) stretching from \( \theta = 0 \) to \( \theta = 2\pi \). Each petal goes from outward at \( r = 4 \) to inward at \( r = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Polar Equations
Graphing polar equations might seem daunting, but with a bit of understanding, it becomes quite manageable. A polar equation is expressed in the form \( r = f(\theta) \), where \( r \) represents the distance from the origin and \( \theta \) denotes the angle from the positive x-axis. These graphs help reveal how \( r \) changes as \( \theta \) varies. To graph a polar equation:
- Start by understanding the functional form, like \( r = 4 \cos^2\left(\frac{1}{2} \theta\right) \), which indicates how the distance changes.
- Calculate key values of \( r \) for different \( \theta \) values to get a sense of the shape.
- Use symmetry to simplify plotting since polar graphs often display symmetrical properties.
- Plot the calculated points and connect them smoothly, keeping the continuity and patterns in mind.
Cosine Function in Polar Coordinates
The cosine function plays a vital role in polar coordinates. Given a function like \( r = 4 \cos^2\left(\frac{1}{2}\theta\right) \), the cosine function affects both the radius \( r \) and the periodicity of the graph. In this context:
- \( \cos^2\left(\frac{1}{2}\theta\right) \) indicates that the cosine function is squared, ensuring all outputs are non-negative, impacting the radial distance from the pole (origin).
- The factor \( \frac{1}{2} \theta \) slows down the frequency of the oscillation, resulting in more pronounced petal shapes on the graph.
- Every increase in \( \theta \) dictates how the cosine function modifies \( r \), tracing a path that reflects the angular progression.
Symmetry in Polar Graphs
Symmetry simplifies the process of plotting polar graphs. It allows predictions about a graph's structure without having to calculate every point. The equation \( r = 4 \cos^2\left(\frac{1}{2}\theta\right) \) is symmetric due to the nature of the cosine function:
- Symmetry across the polar axis makes the graph predictable, meaning the graph looks the same above and below the horizontal line through the pole.
- This symmetry results because \( \cos^2\) is an even function, meaning the graph remains unchanged if \( \theta \) is replaced with \(-\theta\).
- Utilizing symmetry helps simplify plotting by reducing the amount of computation needed.
Polar Roses
Polar roses are elegant curves characterized by petal-like formations in polar plots. The equation \( r = 4 \cos^2\left(\frac{1}{2}\theta\right) \) describes a type of polar rose:
- The squared cosine function naturally creates a rose pattern since the output ranges only from 0 to a maximum positive value, here 4.
- The petal's symmetry and structure are determined by the function's form and periodicity; for instance, \( \frac{1}{2}\theta\) signifies a specific polar rose type with a certain number of petals.
- These roses typically cycle between maximum and minimum values, manifesting beautifully symmetrical curves on the polar graph.
