Problem 71 - Precalculus

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(r=10 \sin 5 \theta\) is a rose curve with five petals.

Step-by-Step Solution

Answer

The statement is true. The graph of the equation \(r=10 \sin 5 \theta\) is indeed a rose curve with five petals.

Step 1 : Define the Equation

The given polar equation is \(r=10 \sin 5 \theta\). This is a sine function with scaling factor \(a=10\) and \(n=5\).

Step 2: Identify the Number of Petals

The value of \(n\) is 5, which is an odd number. When \(n\) is odd in a rose curve function \(r=a \sin (n \theta)\), the number of petals is equal to \(n\). Therefore, the graph of this equation should have 5 petals.

Step 3: Verify the Statement

The statement says that the graph of the equation is a rose curve with 5 petals. Our analysis shows that this is indeed the case. Hence, the statement is true.

Key Concepts

Rose Curve

A rose curve is a fascinating type of graph that forms beautiful petal-like shapes. When dealing with polar coordinates, a rose curve arises from equations of the form \( r = a \sin(n \theta) \) or \( r = a \cos(n \theta) \). Here, \(a\) represents the amplitude which determines the length of the petals, and \(n\) affects the number of petals.

• If \(n\) is odd, the curve will have \(n\) petals.
• If \(n\) is even, the curve will have \(2n\) petals.

In the problem, we have \( r=10 \sin 5 \theta \), where \( n = 5 \). As 5 is odd, the graph will produce 5 petals, confirming the statement given in the exercise.

Sine Function

The sine function is one of the fundamental trigonometric functions, and it plays a crucial role in defining rose curves in polar equations. Its general form in polar coordinates is \( r = a \sin(n \theta) \). The sine function oscillates between -1 and 1, influencing how the radius \(r\) behaves, particularly how it stretches, compresses, or even reverses between the poles of the coordinate system.
In a rose curve like \( r = 10 \sin 5 \theta \), the sine function dictates the symmetrical petals' formation as \(\theta\) changes. It ensures the curve is centered around the origin, expanding outward and inward continuously as the value of \(\theta\) rotates the point, following the oscillation pattern of \(\sin(n \theta)\).

Graph of a Polar Equation

The graph of a polar equation provides a distinct way to visualize functions compared to Cartesian coordinates. Polar graphs plot points based on a distance \(r\) from the origin and an angle \(\theta\). This method is especially helpful for equations like \( r = 10 \sin 5 \theta \), where the graphical representation forms a complex pattern such as a rose curve.

• The distance \(r\) depends on both the sine function and angle \(\theta\), dynamically changing as \(\theta\) increases or decreases.
• The angle \(\theta\) dictates how the graph rotates in a circular motion.

Drawing the graph involves plotting the calculated points for several values of \(\theta\), thereby revealing the rose curve's petal-like structure of the equation. With \( r = 10 \sin 5 \theta \), the process captures the beauty of its five distinct petals as expected.