/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 Sketch the graph of the polar eq... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 65

Sketch the graph of the polar equation. $$r=3 \cos \theta$$

Short Answer

Expert verified
The graph of \( r = 3\cos\theta \) is a circle centered at (1.5, 0) with radius 1.5.

Step by step solution

01

Understand the polar equation form

The given equation is in the form of \( r = a \cos \theta \), where \( a = 3 \). This is a specific type of polar equation known as a lima莽on, and when \( a \) is positive, the graph is symmetric about the polar axis (the horizontal axis in a polar coordinate system).
02

Identify key points

To sketch the graph, determine the values of \( r \) for standard angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).- When \( \theta = 0 \), \( r = 3\cos(0) = 3 \).- When \( \theta = \frac{\pi}{2} \), \( r = 3\cos(\frac{\pi}{2}) = 0 \).- When \( \theta = \pi \), \( r = 3\cos(\pi) = -3 \). Since \( r \) is negative, this point is plotted at \( 3 \) units in the opposite direction.- When \( \theta = \frac{3\pi}{2} \), \( r = 3\cos(\frac{3\pi}{2}) = 0 \).- When \( \theta = 2\pi \), \( r = 3\cos(2\pi) = 3 \).
03

Sketch the graph

Using the key points calculated, the graph can be plotted. Begin by drawing the entire pole (origin) and the radial line for \( \theta = 0 \). Plot the point at \( 3 \) along this line. Then plot the points at \( 0 \), \(-3\), and \( 0 \) at \( \theta = \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) respectively.The graph will be a circle with a radius of 1.5 centered at \( (1.5, 0) \) in the polar coordinate plane, corresponding to a circle with center \( (1.5, 0) \) and radius \( 1.5 \) in Cartesian coordinates.

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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 unique type of curve that can be defined using polar coordinates. The name "lima莽on" comes from the French word for "snail," aptly describing its snail-like shape.
  • It is defined by the polar equation form: \( r = a + b \cos(\theta) \) or \( r = a + b \sin(\theta) \).
  • The appearance of a lima莽on can vary widely depending on the relation between the constants \(a\) and \(b\).
  • When \( b = a \), the lima莽on takes the form of a cardioid.
In the specific case of \( r = 3 \cos(\theta) \), the polar equation resembles a lima莽on with only a cosine term. This indicates the symmetry about the polar axis, as well as how the curve will appear relative to the circle centered at the pole. Understanding how the parameters influence the shape is crucial as it determines whether the lima莽on will have an inner loop, be dimpled, or appear convex.
Polar Equation Graphing
Polar equation graphing involves plotting points in a coordinate system that uses the radius and angle, rather than the traditional x and y Cartesian coordinates. Here's how to graph such equations effectively:
  • Start with the given polar equation, such as \( r = 3 \cos(\theta) \).
  • Choose standard angles鈥攖hese are usually \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).
  • Use the equation to calculate the radius \( r \) for each of these angles.
After these calculations, plot the points. For example, when \( \theta = 0 \) and \( r = 3 \cos(0) \), the point is plotted three units away from the origin along the positive x-axis. For \( \theta = \frac{\pi}{2} \), \( r \) equals 0, meaning the point remains at the origin. This approach helps connect the dots visually, creating curves like circles or lima莽ons easily represented in polar coordinates.
Cosine Function
The cosine function is a fundamental trigonometric function that was crucial in plotting the given polar equation, \( r = 3 \cos(\theta) \). This function oscillates between -1 and 1 as \( \theta \) varies from 0 to \( 2\pi \).
  • It is important to understand how the cosine function modifies the radius \( r \) across different angles.
  • The maximum value of \( \cos(\theta) \) is 1, occurring at \( \theta = 0, 2\pi \), contributing the positive maximum radius of 3 in this example.
  • At \( \theta = \pi \),\( \cos(\theta) \) is -1, reflecting r=-3, but in polar terms, this represents a point at 3 units opposite to the angle's direction.
This oscillation forms the characteristic shape of the graph in polar coordinates. Understanding how cosine influences the radial distance from the pole to these points is essential to drawing precise curves and interpreting their behavior properly.

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Most popular questions from this chapter

Exer. \(67-68\) : The planets move around the sun in elliptical orbits. Given the semimajor axis a in millions of kilometers and eccentricity \(e,\) graph the orbit for the planet. Center the major axis on the \(x\) -axis, and plot the location of the sum at one focus. $$\text { Pluto's path } a=5913, \quad e=0.249$$

Shown in the figure is the Lissajous figure given by $$x=2 \sin 3 t, \quad y=3 \sin 1.5 t, \quad t \geq 0$$ Find the period of the figure-that is, the length of the smallest \(t\) -interval that traces the curve. CAN'T COPY THE GRAPH

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