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Chapter 10: Problem 39

Identify and graph each polar equation. $$ r=2+2 \cos \theta $$

Short Answer

Expert verified
Graph the 濒颈尘补莽辞苍 with an inner loop present at \( r = 2 + 2 \, \cos \theta \).

Step by step solution

01

Recognize the Standard Polar Form

Identify the given polar equation and recognize it as part of a standard form. The equation given is \[ r = 2 + 2 \, \cos \theta \]This is a polar equation that can be identified as a 濒颈尘补莽辞苍.
02

Determine Key Characteristics

A 濒颈尘补莽辞苍 of the form \[ r = a + b \, \cos \theta \]The constants in the equation are: \[ a = 2 \, \text{and} \, b = 2 \]Since \[ a = b \]this specific 濒颈尘补莽辞苍 will have an inner loop.
03

Find Important Points

Find and plot the points for specific values of \( \theta \):For \( \theta = 0 \):\[ r = 2 + 2 \, \cos(0) = 4 \]For \( \theta = \frac{\pi}{2} \):\[ r = 2 + 2 \, \cos \left( \frac{\pi}{2} \right) = 2 \]For \( \theta = \pi \):\[ r = 2 + 2 \, \cos(\pi) = 0 \]For \( \theta = \frac{3\pi}{2} \):\[ r = 2 + 2 \, \cos \left( \frac{3\pi}{2} \right) = 2 \]
04

Sketch the Graph

Using the points calculated from Step 3, sketch the graph of the 濒颈尘补莽辞苍. The plot should reflect the shape with an inner loop, starting at (4, 0) and going through (0, 蟺).

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Key Concepts

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

濒颈尘补莽辞苍
A 濒颈尘补莽辞苍 is a type of curve represented by polar equations of the form either \( r = a + b \, \text{cos} \, \theta \) or \( r = a + b \, \text{sin} \, \theta \). Lima莽ons have unique and interesting shapes that vary depending on the values of \( a \) and \( b \). In our case, \( r = 2 + 2 \, \text{cos} \, \theta \), the constants \( a \) and \( b \) both equal 2. Some key characteristics of 濒颈尘补莽辞苍s include:
  • If \( a < b \), the 濒颈尘补莽辞苍 has an inner loop.
  • If \( a = b \), it's a special 濒颈尘补莽辞苍 with an inner loop.
  • If \( a > b \), it appears without an inner loop.
In our problem, since \( a = b \), the resulting 濒颈尘补莽辞苍 possesses an inner loop. When you plot this curve, you'll notice how it loops back towards the origin.
graphing polar coordinates
Understanding polar coordinates is essential for graphing polar equations like the 濒颈尘补莽辞苍. A polar coordinate system represents a point in the plane using the distance from the origin \( r \) and the angle \( \theta \) from the positive x-axis. Here鈥檚 a quick overview:
  • The distance \( r \) measures how far away a point is from the origin (pole).
  • The angle \( \theta \) measures the rotational offset from the positive x-axis (polar axis).
To graph our polar equation \( r = 2 + 2 \, \text{cos} \, \theta \):
  • Calculate \( r \) for several values of \( \theta \), like \( 0 \, \text{and} \, \frac{\text{蟺}}{2}, \, \text{蟺}, \, \frac{\text{3蟺}}{2} \).
  • Plot these points on the polar coordinate plane.
  • Connect the points to visualize the 濒颈尘补莽辞苍 curve.
This process lets you see the loop and inner structures by plotting key points where \( \theta \) equals 0, \( \frac{\text{蟺}}{2} \), 蟺, and \( \frac{\text{3蟺}}{2} \). Effective graphing results in a comprehensible graph of your polar equation.
trigonometric functions
Trigonometric functions like \( \text{cos} \, \theta \) and \( \text{sin} \, \theta \) are fundamental to understanding polar equations. In our exercise, \( r = 2 + 2 \, \text{cos} \, \theta \), we rely on the cosine function to determine \( r \) for different \( \theta \) values. Let's break down their role:
  • Cosine function \( \text{cos} \, \theta \) returns a value between -1 and 1.
  • For \( \theta = 0 \), \( \text{cos} \, 0 = 1 \), effectively making \( r = 2 + 2 \times 1 = 4 \).
  • For \( \theta = \frac{\text{蟺}}{2} \), \( \text{cos} \, \frac{\text{蟺}}{2} = 0 \), leading to \( r = 2 + 2 \times 0 = 2 \).
  • For \( \theta = \text{蟺} \), \( \text{cos} \, \text{蟺} = -1 \), resulting in \( r = 2 + 2 \times -1 = 0 \).
Understanding cosine values at these key angles lets you plot points easily. Consequently, making the graphing of polar equations clear and straightforward.

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