Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given polar equation and verify its symmetry (see Examples \(1-3)\). \(r=5 \cos 3 \theta\) (three-leaved rose)

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 5 \cos 3\theta\) is a three-leaved rose symmetric about the x-axis, origin, and the line \(\theta = \frac{\pi}{3}\).

Step 1: Understand the Polar Equation

The given equation is in the form \(r = 5 \cos 3\theta\). This represents a polar curve known as a 'three-leaved rose'. The value of \(3\theta\) suggests that the curve will have three petals. The number of petals is determined by the coefficient of \(\theta\), which is odd in this case.

Step 2: Identify Key Points

To sketch the graph, identify key points by substituting specific \(\theta\) values. Start with \(\theta = 0\), \(\theta = \frac{\pi}{6}\), \(\theta = \frac{\pi}{3}\), \(\theta = \frac{\pi}{2}\), and continue until \(\theta = \pi\). Calculate \(r\) for each angle: - \(\theta = 0\), \(r = 5\) - \(\theta = \frac{\pi}{6}\), \(r = 2.5\) - \(\theta = \frac{\pi}{3}\), \(r = 0\) - \(\theta = \frac{\pi}{2}\), \(r = -5\) - \(\theta = \pi\), \(r = -5\) - Continue to fill more values to see the repetition every \(\frac{2\pi}{3}\) radians.

Step 3: Plot Points and Sketch the Graph

With the identified key points, plot them on polar coordinate paper. Connect the points smoothly respecting the curve's petal pattern. Remember, the cosine function also implies a symmetric behavior about the polar axis. This symmetry should be visible in the graph as it completes one full period.

Step 4: Check for Symmetry

Examine the graph to verify symmetry. A rose curve of the form \(r = a \cos(n\theta)\) with \(n\) being odd exhibits symmetry concerning the x-axis. Here, symmetry about the polar axis, the origin, and the line \(\theta = \frac{\pi}{n}\) is expected. Confirm visually that each petal is symmetrically aligned about these axes.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates offer an alternative to Cartesian coordinates for understanding points in a plane. Unlike Cartesian coordinates, which use a grid of perpendicular axes, polar coordinates describe a point based on its distance from a reference point (called the pole) and the angle from a reference direction, typically the positive x-axis.

Key components of polar coordinates include:

• Radial Coordinate (r): This represents the distance from the pole to the point.
• Angular Coordinate (θ): This is the angle formed between the reference direction and the line connecting the point to the pole, measured in radians.

The transformation from Cartesian to polar coordinates is straightforward with the formulas:

\(r = \sqrt{x^2 + y^2}\) and \(θ = \arctan\left(\frac{y}{x}\right)\).

Remember, polar equations often reveal rotational symmetries which are simpler to describe using this coordinate system.

Three-Leaved Rose

The 'three-leaved rose' is an enchanting type of curve represented by polar equations, namely of the form \(r = a \cos n\theta\) or \(r = a \sin n\theta\). These equations create petal patterns on the polar graph, leading to beautiful and symmetric designs.

In our equation, \(r = 5 \cos 3\theta\), '3-leaved rose’ implies that the graph has three loops or petals. This is because the coefficient of \(n=3\) is odd.

Characteristics of the three-leaved rose include:

• Maximum radius occurs when \(\cos 3\theta = 1\), resulting in \(r = a\).
• The petals are symmetrically distributed around the pole.
• Petals appear every \(\frac{2\pi}{n}\) radians.

These rose curves are not just mathematic curiosities; they often serve applications in various fields including physics and engineering due to their elegant symmetry.

Graph Symmetry

Symmetry in graphs is a powerful visual property that helps understand the nature of a function. For polar curves, symmetry can simplify graphing efforts and offer insights into the behavior of the curve.

Symmetry types to look for include:

• Symmetry about the x-axis: For equations in the form \(r = a \cos n\theta\), if you replace \(θ\) with \(-θ\) and find the same equation, the graph is x-axis symmetric.
• Symmetry about the y-axis: Similarly, for \(r = a \sin n\theta\), replace \(θ\) with \(\pi - θ\).
• Symmetry about the origin: If replacing \(r\) and \(θ\) with \(-r\) and \(-θ\) results in the same equation, the graph is symmetric with respect to the origin.

For the three-leaved rose, the graph shows symmetry about the x-axis, proving visually beautiful and mathematically significant characteristics.

Trigonometric Functions

Trigonometric functions are the backbone of many polar equations. They relate angles to ratios of lengths in right triangles and provide a periodic nature, fundamental to describing oscillating systems.

Key trigonometric functions include:

• Sine (\(\sin\)): Models the y-component of a unit circle as the angle sweeps from 0 to \(2\pi\).
• Cosine (\(\cos\)): Represents the x-component of the unit circle and begins its cycle from maximum 1.
• Tangent (\(\tan\)): Describes the slope or steepness of the angle, being \(\frac{\sin θ}{\cos θ}\).

In the polar equation \(r=5 \cos 3\theta\), cosine determines the radial variation, creating the petals of the rose by adjusting \(r\)'s magnitude based on \(\theta\). Trigonometric identities and properties assist with identifying periodic behaviors and symmetries intrinsic to the graph.