Question

Find the symmetry of the rose defined by the equation \(r=3 \sin (2 \theta)\) and create a graph.

Short answer

The rose curve \( r = 3 \sin(2\theta) \) is symmetric about the origin and has 4 petals.

Step-by-step solution

Understanding the Polar Equation

The given equation is in polar form: \[ r = 3 \ sin(2 \theta) \] This represents a rose curve commonly observed in polar graphs. The coefficient within the sine function, 2, is significant in determining symmetry and other properties of the curve.

Identify the Symmetry

To find the symmetry of the rose curve, we analyze the trigonometric component. The graph of \( r = 3 \sin(2 \theta) \) is symmetric with respect to the origin. This is because for each \( \theta \) there exists an angle \( \theta + \pi \) for which \(\sin(2 (\theta + \pi)) = \sin(2 \theta)\). This symmetry is due to the even nature of \( \sin(2 \theta)\). Therefore, the curve is symmetric about the origin.

Determine Number of Petals

The number of petals of a rose curve \( r = a \sin(n \theta) \) depends on \( n \). If \( n \) is even, it will have \( 2n \) petals. Since in our equation \( n = 2 \), the rose will have \( 2 \times 2 = 4 \) petals.

Choose Points to Plot

Choose specific values of \( \theta \) to plot points, making sure to capture the entire cycle of \( 0 \) to \( 2\pi \). For example, choose \( \theta \) values like \( 0, \pi/4, \pi/2, \pi, 3\pi/2, 2\pi \) and calculate corresponding \( r \) values using the equation.

Plot the Points and Draw the Curve

Calculate the points for chosen \( \theta \) values: - At \( \theta = 0 \), \( r = 0 \) - At \( \theta = \pi/4 \), \( r = 3 \sin(\pi/2) = 3 \)- At \( \theta = \pi/2 \), \( r = 0 \)Repeat this calculation for other \( \theta \) values to get full shape. Plot these points on polar graph paper and connect them to complete the symmetry with 4 petals.

Key concepts

Symmetry in Polar Graphs

Understanding symmetry in polar graphs is crucial when analyzing curves like rose curves. Symmetry refers to the balanced and identical arrangement of a graph around a central point or line. In polar coordinates, we often look for the graph's symmetry with respect to the polar axis (x-axis), the line \(\theta = \frac{\pi}{2}\), or the origin.
This exercise deals with the equation \(r = 3 \sin(2 \theta)\), which shows symmetry about the origin. This type of symmetry means that for every point \( \theta \), there is a corresponding point at \( \theta + \pi \) that maps to an equivalent value of \( r \), reflecting the curve in all quadrants. Such symmetry results from the periodic nature of trigonometric functions used in polar equations.

• Symmetry about origin often suggests an even number of petals in rose curves.
• Check through substitution of \( \theta \) with \( \theta + \pi \).

This property is crucial in predicting the graph's structure and allows simplification in plotting polar curves.

Rose Curves

Rose curves are intriguing and beautiful patterns observed in polar graphs. These curves are made using equations of the form \( r = a \sin(n \theta) \) or \( r = a \cos(n \theta) \). Here, \( a \) defines the petal length, and \( n \) determines the number of petals.

• If \( n \) is even, the curve displays \( 2n \) petals.
• If \( n \) is odd, the curve exhibits \( n \) petals.

In the given problem, \( r = 3 \sin(2 \theta) \) implies that \( n = 2 \). Consequently, the curve presents \( 4 \) petals because when \( n \) is even, the number of petals equals \( 2n = 4 \).
This characteristic of rose curves provides insight into their formation and how they visually appear on polar coordinate graphs. Recognizing these features aids in drawing or identifying such curves.

Trigonometric Functions

Trigonometric functions like sine and cosine play a pivotal role in defining polar equations. In our problem, the trigonometric function used is \( \sin(2 \theta) \). Trigonometric functions are periodic, meaning they repeat values in a consistent pattern over intervals, influencing the nature and layout of the graph.
For \( \sin(2 \theta) \), the period is \( \pi \), meaning every \( \pi \) radians, the function repeats itself. This periodicity is integral when determining the curve's layout, such as the symmetry and number of petals in a rose curve.

• Sine functions can create symmetrical and smooth loops in polar equations.
• Understanding the period helps in predicting graph transitions.

Mastering these aspects simplifies graph plotting and enables students to predict how changes in \( n \) or \( a \) might alter the graph's appearance, providing a foundation for deeper exploration in higher mathematics.