Problem 61 How many petals does the polar r... [FREE SOLUTION]

Chapter 11: Problem 61

How many petals does the polar rose \(r=\sin (2 \theta)\) have? What about \(r=\sin (3 \theta), r=\sin (4 \theta)\) and \(r=\sin (5 \theta) ?\) With the help of your classmates, make a conjecture as to how many petals the polar rose \(r=\sin (n \theta)\) has for any natural number \(n\). Replace sine with cosine and repeat the investigation. How many petals does \(r=\cos (n \theta)\) have for each natural number \(n ?\)

Short Answer

Expert verified

For \( r = \sin(n\theta) \) and \( r = \cos(n\theta) \): if \( n \) is even, there are \( 2n \) petals; if \( n \) is odd, there are \( n \) petals.

Step by step solution

Understanding the Polar Rose Equation

The polar equation for a rose, generally of the form \( r = \text{function}(n\theta) \), describes a curve with petal-like shapes. Here, we explore how the value of \( n \) affects the number of petals for both sine and cosine functions.

Analyzing r=sin(2胃)

For \( r = \sin(2\theta) \), the rose curve will have \( 2n \) petals because the coefficient \( n \) is 2. Thus, the equation \( r = \sin(2\theta) \) results in 4 petals.

Analyzing r=sin(3胃)

For \( r = \sin(3\theta) \), since \( n \) is odd, there will be \( n \) petals. Therefore, \( r = \sin(3\theta) \) will produce a polar rose with 3 petals.

Analyzing r=sin(4胃)

For \( r = \sin(4\theta) \), \( n \) is even, leading to \( 2n \) petals. Thus, the polar rose \( r = \sin(4\theta) \) will have 8 petals.

Analyzing r=sin(5胃)

For \( r = \sin(5\theta) \), \( n \) is odd, and as such, there will be \( n \) petals. Hence, the equation \( r = \sin(5\theta) \) results in 5 petals.

Conjecture for r=sin(n胃)

For a polar rose given by \( r = \sin(n\theta) \), if \( n \) is even, the rose has \( 2n \) petals, and if \( n \) is odd, it has \( n \) petals.

Analyzing r=cos(2胃)

Just like \( r = \sin(2\theta) \), for the cosine function, \( r = \cos(2\theta) \) also results in \( 2n \) petals, giving us 4 petals.

Analyzing r=cos(3胃)

For \( r = \cos(3\theta) \), since \( n \) is odd, the rose has \( n \) petals. Therefore, \( r = \cos(3\theta) \) results in 3 petals.

Analyzing r=cos(4胃)

For \( r = \cos(4\theta) \), \( n \) is even, leading to \( 2n \) petals. Thus, \( r = \cos(4\theta) \) will have 8 petals.

Analyzing r=cos(5胃)

With \( r = \cos(5\theta) \) where \( n \) is odd, the number of petals is \( n \). Therefore, this results in 5 petals.

Conjecture for r=cos(n胃)

For \( r = \cos(n\theta) \), the rule is similar to the sine function: if \( n \) is even, expect \( 2n \) petals, whereas if \( n \) is odd, expect \( n \) petals.

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Key Concepts

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

Petal Count

Understanding the petal count of a polar rose is key to predicting its appearance. A polar rose is a type of graph represented by polar equations with sine and cosine functions involved. The number of petals in a polar rose depends on the coefficient of \(\theta\) in the equation \(r = \sin(n\theta)\) or \(r = \cos(n\theta)\):

If \(n\) is even, the rose has \(2n\) petals.
If \(n\) is odd, the rose has \(n\) petals.

This rule applies to both sine and cosine functions. Remembering this simple pattern helps in analyzing any polar rose equation and predicting its petal count accurately.

Polar Equations

Polar equations provide a unique way to express curves with a radial (distance) and angular component rather than the usual x-y Cartesian coordinates. A basic polar equation for a rose takes the form \(r = \text{function}(n\theta)\), where:

\(r\) is the radius, the distance from the pole (origin)
\(\theta\) is the angle measured from the positive x-axis
\(n\) affects the number of oscillations or petals

Polar roses are a particular type of polar equation known for their aesthetic, petal-like patterns, which differ based on the value of \(n\). This elegant mathematical representation is why polar equations are favored in fields requiring circular or periodic symmetry.

Sine and Cosine Functions

The sine and cosine functions are foundational in graphing polar roses. In these equations, they oscillate as angles change, creating petal-shaped graphs unique to these periodic functions. Here are key points:

The function \(\sin(n\theta)\) and \(\cos(n\theta)\) define the radial distances for each angle \(\theta\).
Sine and cosine functions start their periods at different points. Sine begins at zero, while cosine starts at maximum radius.
These differences affect the rotation and orientation of roses, but the petal count remains governed by the coefficient \(n\).

Understanding these functions' behaviors helps explain why and how each polar rose forms in its distinct manner.

Even and Odd Coefficients

Even and odd coefficients in polar equations decide the symmetry and petal count of a polar rose. This is one of the most fascinating aspects of these mathematical expressions. Consider these points for clarity:

When \(n\) is even in \(r = \sin(n\theta)\) or \(r = \cos(n\theta)\), the polar rose exhibits \(2n\) symmetric petals.
If \(n\) is odd, the rose graph displays \(n\) petals, typically with a petal "at the pole."
These behaviors highlight the periodic nature of trigonometric functions and how their symmetries influence graph shapes.

Recognizing the effects of even and odd \(n\) allows for immediate predictions of the graph's form, making graph-sketching and understanding much easier.

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Short Answer

Step by step solution

Understanding the Polar Rose Equation

Analyzing r=sin(2胃)

Analyzing r=sin(3胃)

Analyzing r=sin(4胃)

Analyzing r=sin(5胃)

Conjecture for r=sin(n胃)

Analyzing r=cos(2胃)

Analyzing r=cos(3胃)

Analyzing r=cos(4胃)

Analyzing r=cos(5胃)

Conjecture for r=cos(n胃)

Key Concepts

Petal Count

Polar Equations

Sine and Cosine Functions

Even and Odd Coefficients

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