Problem 57 - Precalculus

Question

Graph the polar equation. \(r=4 \sin (5 \theta)\)

Step-by-Step Solution

Answer

The polar graph is a rose curve with 5 petals, each having a length of 4.

Step 1: Understand the Equation

The given polar equation is \( r = 4 \sin(5\theta) \). This is a polar equation where \( r \) is a function of \( \theta \). The equation is in the form \( r = a \sin(n\theta) \), which typically represents a rose curve.

Step 2: Identify the Type of Graph

The equation \( r = a \sin(n\theta) \) represents a rose curve. When \( n \) is an odd integer, the rose has \( n \) petals; when \( n \) is even, the rose has \( 2n \) petals. Here, \( n = 5 \), which is an odd integer so the graph will have 5 petals.

Step 3: Determine the Petal Length

The length of each petal is determined by \( a \), which in this case is 4. Thus, each petal of the rose curve will extend to a maximum radius of 4 from the pole.

Step 4: Sketch the Polar Graph

Plot the rose curve with 5 petals on polar coordinates. Each petal starts at the origin and extends outward, with a maximum radius of 4. The petals are evenly spaced around the origin with angles incrementing by \(\frac{2\pi}{5}\) between successive petals.

Key Concepts

Rose Curve

A rose curve is a very interesting type of polar graph. It's called a rose curve because its shape resembles the petals of a rose flower. These curves are expressed with polar equations that have a specific format: \(r = a \sin(n\theta)\) or \(r = a \cos(n\theta)\). The number of petals that a rose curve has depends on the value of \(n\) in the equation.

• If \(n\) is an odd number, the rose curve will have exactly \(n\) petals.
• If \(n\) is even, the rose curve will display \(2n\) petals.

In our specific example \(r = 4 \sin(5\theta)\), the number 5 is odd, indicating the curve will blossom 5 petals. Another important factor is the length of the petals, determined by the coefficient \(a\). It indicates how far the petals extend from the center, or origin, of the polar grid. Here, \(a = 4\), so each petal reaches a radius of 4.

Polar Coordinates

Polar coordinates provide a way to represent points in a plane, different from the traditional Cartesian coordinates, which use \(x\) and \(y\) values. In polar coordinates, a point is represented by \((r, \theta)\), where:

• \(r\) is the radial distance from the origin (or pole).
• \(\theta\) is the angle measured from the positive x-axis in radians.

This system is especially useful for plotting complex curves, like rose curves, because of its natural ability to accommodate radial patterns and symmetries. When using polar coordinates, every point is described in terms of its distance from a fixed central point and the angle from a fixed direction. This can simplify the plotting of certain graphs compared to Cartesian coordinates.

Graphing Polar Equations

Graphing polar equations involves plotting points on a polar grid based on their polar coordinates. Each point is represented by \((r, \theta)\), where the distance \(r\) from the origin and the angle \(\theta\) determine its position. Here are some steps to graph polar equations effectively:

• Identify critical points by evaluating \(r\) at various angles \(\theta\)
• Determine symmetries or predict behaviors from the polar equation itself.
• Plot each point \((r, \theta)\) accurately on the polar grid.
• Sketch smooth curves connecting the points, repeating patterns where applicable.

For our rose curve \(r = 4 \sin(5\theta)\), begin by plotting key points as \(\theta\) changes. The maximum point for each petal will occur when \(\sin(5\theta) = 1\), resulting in \(r = 4\). Each petal forms in sequence every \(\frac{2\pi}{5}\) radians, completing the beautiful pattern of the rose on the grid.