/*! 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 44 Sketch the graph of each polar e... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 44

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

Short Answer

Expert verified
The graph is a circle centered at (1.5, 0) with radius 1.5.

Step by step solution

01

- Understand the Equation

Given the polar equation \( r = 3 \cos \theta \), recognize that this is a polar equation involving the cosine function.
02

- Identify the Type of Graph

Notice that \( r = 3 \cos \theta \) is a polar equation representing a circle. This can be understood because the polar form \( r = a \cos \theta \) represents a circle centered on the polar axis.
03

- Determine the Radius and Position

Here, the coefficient 3 indicates that the maximum value of \( r \) is 3. Therefore, the radius of the circle is 3, and it is centered at \( (1.5, 0) \) since for \( r = a \cos \theta \), the center is at \( (a/2, 0) \).
04

- Plot Key Points

Calculate key points by choosing values for \( \theta \) to find corresponding \( r \) values. For example, when \( \theta = 0 \), \( r = 3 \). When \( \theta = \frac{\pi}{2} \), \( r = 0 \). When \( \theta = \pi \), \( r = -3 \) which translates to \( r = 3 \) in the opposite direction.
05

- Draw the Circle

Using the points and knowing the general shape, draw the circle centered at \( (1.5, 0) \) with a radius of 1.5.

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Key Concepts

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

cosine function
The cosine function is a fundamental mathematical function often denoted as \( \cos\theta \) \theta represents an angle, usually in radians or degrees. \ The cosine function is periodic, meaning it repeats its values at regular intervals. This period is \( 2\pi \) for angles in radians. Cosine values range from -1 to 1, creating a wave known as the cosine wave. In the context of polar equations like \( r = 3 \cos\theta \), the cosine function can describe circular shapes. When multiplying cosine by a constant, like 3 in this equation, the value scales the resultant radius. Hence, the maximum and minimum radii depend on the multiplier used.

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Most popular questions from this chapter

Convert the polar coordinates of each point to rectangular coordinates. $$ \left(\sqrt{3}, 100^{\circ}\right) $$

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