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Chapter 13: Problem 27

Draw a sketch of the graph of the given equation.\(r=4-3 \cos \theta\) (lima莽on)

Short Answer

Expert verified
Plot points for specific \(\theta\) values and draw a smooth curve. Example points: (1, 0), (4, \(\frac{\pi}{2}\)), (7, \(\pi\)).

Step by step solution

01

Understand the Given Equation

The given equation is in polar coordinates: \(r=4-3 \cos \theta\). This is an equation of a lima莽on. 尝颈尘补莽辞苍s are a type of polar curve which can have different shapes, depending on the parameter values.
02

Identify Key Points

Calculate the values of \(r\) for key angles \(\theta\). For example: \(\theta = 0\): \(r = 4 - 3 \cos(0) = 4 - 3 = 1\) \(\theta = \frac{\pi}{2}\): \(r = 4 - 3 \cos(\frac{\pi}{2}) = 4 - 3 \cdot 0 = 4\) \(\theta = \pi\): \(r = 4 - 3 \cos(\pi) = 4 - 3 \cdot (-1) = 4 + 3 = 7\)Repeat for other important angles if needed.
03

Plot Key Points

Plot the calculated points on polar coordinate paper or use a Cartesian coordinate system in a polar context. For example, plot (1, 0), (4, \(\frac{\pi}{2}\)), and (7, \(\pi\)) on the polar graph.
04

Draw the 尝颈尘补莽辞苍 Curve

Connect the points smoothly, considering the symmetry and shape of a lima莽on. It usually has an inner loop if the magnitude of the coefficients results in one. The form 4 - 3 cos(胃) typically will not have a loop since 4 > 3.

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Key Concepts

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

尝颈尘补莽辞苍
A lima莽on is a type of polar graph defined by equations of the form \( r = a + b \, \cos(\theta) \) or \( r = a - b \, \cos(\theta) \). Depending on the values of \( a \) and \( b \), the shape of the lima莽on can vary significantly. Some common variations include:
  • When \( a = b \), the curve features an inner loop.
  • When \( a > b \), the curve resembles a dimpled shape without an inner loop.
  • If \( b > a \), the lima莽on will have an outer loop.
The given equation, \( r = 4 - 3 \, \cos(\theta) \), falls under the category where \( a > b \), indicating that the lima莽on will not have an inner loop but will instead have a dimpled heart-like shape. Observing how varying the values of \( a \) and \( b \) affects the curve's structure helps understand different forms of lima莽ons better.
Polar Graph
A polar graph represents data points in terms of the polar coordinate system, where each point is determined by a radius \( r \) and an angle \( \theta \). Unlike Cartesian coordinates that use \( (x, y) \) pairs to locate points, polar coordinates use \( (r, \theta) \) pairs.

Here's how to understand the polar graph:
  • \( r \): The distance from the origin to the plotted point.
  • \( \theta \): The angle measured in radians or degrees from the positive x-axis (counter-clockwise).
For our lima莽on equation, we calculated the radius \( r \) for various angles \( \theta \). These angles form key points that help draw the overall shape of the curve. Key points like \( (1, 0) \), \( (4, \frac{\pi}{2}) \), and \( (7, \pi) \) provide essential anchors to sketch the polar graph accurately.
Key Angles
Key angles play a crucial role in mapping the graph of a polar equation. For complex curves like the lima莽on, computing the radius \( r \) at specific angles helps in plotting critical points accurately. These angles are generally:
  • \( \theta = 0 \)
  • \( \theta = \frac{\pi}{2} \)
  • \( \theta = \pi \)
  • \( \theta = \frac{3\pi}{2} \)
  • Any additional angles that define significant points on the curve
For the equation \( r = 4 - 3 \, \cos(\theta) \), the computed radii for these angles helped in locating the key points (1, 0), (4, \frac{\pi}{2}), and (7, \pi). Once these points were plotted, connecting them smoothly resulted in the dimpled heart-shaped lima莽on, accurately reflecting the polar equation. Measuring radius at key angles ensures that the graph is well-defined and anchors important features of the curve.

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Most popular questions from this chapter

Plot the point having the given set of polar coordinates; then find another set of polar coordinates for the same point for which (a) \(r<0\) and \(0 \leq \theta<2 \pi ;\) (b) \(r>0\) and \(-2 \pi<\theta \leq 0 ;\) (c) \(r<0\) and \(-2 \pi<\theta \leq 0\).\(\left(3, \frac{3}{2} \pi\right)\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=4(1+\sin \theta) \\ r(1-\sin \theta)=3\end{array}\right.\)

Find a polar equation of the graph having the given cartesian equation.\(\left(x^{2}+y^{2}\right)^{2}=4\left(x^{2}-y^{2}\right)\)

Draw a sketch of the graph of the given equation.\(r=2 \cos 4 \theta\) (eight- leafed rose)

Draw a sketch of the graph of the given equation.\(r=4-4 \cos \theta\) (cardioid)
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