Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 46-49, use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. \(r=\sin (5 \theta / 7)\)

Step-by-Step Solution

Answer

Graph the equation in polar mode from \(\theta = 0\) to \(\theta = 4\pi\).

Step 1: Understand the Type of Equation

We have the polar equation \( r = \sin\left(\frac{5\theta}{7}\right) \). Polar equations express the relationship between the radius \( r \) and the angle \( \theta \) in a polar coordinate system.

Step 2: Determine the Interval for \(\theta\)

For trigonometric polar equations, it is often sufficient to choose the interval \(\theta \in [0, 2\pi] \) or larger to capture the full pattern of the curve. Since the equation uses a fraction \( \frac{5\theta}{7} \), consider extending the interval to \([0, 4\pi]\) for a complete illustration.

Step 3: Set Up Graphing Parameters

Using a graphing calculator or computer software, set the parameter \(\theta\) to vary between 0 and \(4\pi\) to ensure the entire curve is graphed. Input this setting to your graphing tool.

Step 4: Graph the Equation

Enter the equation \( r = \sin\left(\frac{5\theta}{7}\right) \) into the graphing calculator or computer software. Ensure that the graph is set to polar mode.

Step 5: Observe the Graph

After plotting the graph, observe the pattern. Typically with such polar equations, you might observe a multi-petal rose-like pattern. Verify that the entire pattern is visible and critiqued using the chosen interval.

Key Concepts

Graphing Calculators

When it comes to visualizing complex equations such as polar equations, graphing calculators are invaluable tools. They allow you to input polar equations directly, enabling a quick and accurate representation of complicated curves. This can be especially helpful when dealing with trigonometric functions or fractional coefficients. Here's a simple guide on how to make the most out of your graphing calculator:

• Ensure your calculator is set to "polar" mode. This is crucial since the default setting is often the Cartesian system, which deals with x and y coordinates instead of polar coordinates.
• Next, enter the polar equation in the format your calculator recognizes, such as r = sin(5θ/7). Look for dedicated functions for sine and theta to make this easier.
• Set the range for θ. To capture intricate patterns or complete cycles, using an extended interval like [0, 4π] is beneficial.
• Finally, graph the equation and observe the pattern. Most graphing calculators have a zoom feature which can be useful for viewing details or getting an overview of the entire polar plot.

Graphing calculators simplify the process of solving and exploring mathematical equations, making learning more interactive and visually engaging.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in understanding polar equations. They describe the ratios of the sides of right triangles and extend into circular functions, which are pertinent in polar coordinates. This section focuses on sine function as used in polar equations:

• The sine function operates over an interval, producing a wave-like pattern that repeats every 2π radians. This makes it ideal for modeling periodic behavior such as oscillations.
• In the polar equation r = sin(5θ/7), the sine function determines how the radius r changes as θ varies. This results in a pattern that can sometimes appear as a rose curve with multiple petals.
• Understanding how fractional coefficients, like 5/7, affect the trigonometric function is crucial. They modify the period and frequency of the sine wave, thus altering the number of loops or petals in the graph.
• Always try to visualize the trigonometric functions' behavior by considering key points such as maxima, minima, and intersections with the axis.

Recognizing these attributes allows for a deeper comprehension of how trigonometric functions define complex polar equations.

Polar Coordinates

Polar coordinates offer a unique method for graphing complex curves, using a system based on a radius and angle relative to a central point. In contrast to the Cartesian coordinate system which uses x and y axes, the polar system provides a different perspective:

• Polar equations consist of r (the radius) and θ (the angle), which jointly determine the position of a point in a plane. Here, the radius can represent distance from the origin, and the angle indicates direction.
• Such a system is especially useful for curves that are naturally circular or spiral, simplifying their representation compared to Cartesian graphs.
• To plot a polar curve, understand that every point is described by an angle and a distance; this is depicted as r = f(θ). Varying θ sketches out different points and shapes.
• When graphing polar equations like r = sin(5θ/7), consider how changes in θ influence the curve's shape. Larger intervals ensure the complete shape is visible, often revealing intricate patterns.

Mastering polar coordinates adds to your mathematical toolkit, offering insight into a world of radial symmetry and elegant curves.