Problem 37 - Precalculus

Question

Use a graphing utility to graph the rotated conic. $$r=\frac{6}{2+\sin (\theta+\pi / 6)}$$

Step-by-Step Solution

Answer

The given function \(r=\frac{6}{2+\sin (\theta+\pi /6)}\) is a polar function describing a rotated conic section. The function creates a conic section together with a rotation by \(\pi /6\) units. The exact shape should be viewed using a graphing utility.

Step 1: Identify Equation Type and Rotation

The given function \(r=\frac{6}{2+\sin (\theta+\pi /6)}\) is a polar function describing a rotated conic section, often either an ellipse or a hyperbola. The rotation here comes from the \(\theta+\pi / 6\) in the sin function. This indicates that the function is shifted \(\pi /6\) units to the left.

Step 2: Plot using a graphing utility

Now that we know what the function represents, we can use a graphing utility tool to plot this function. Input the function \(r=\frac{6}{2+\sin (\theta+\pi /6)}\) into the graphing tool, and set the mode to polar.

Step 3: Interpret the graph

Once the graph has been plotted, we can interpret the shape and rotation. The function should create a conic section rotated by \(\pi /6\) units. The exact shape - a circle, ellipse, or hyperbola - is determined by the behavior of the function.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a radius and an angle. Unlike Cartesian coordinates, which use an x and y axis, polar coordinates use:

• \( r \): the distance from the origin (or pole), and
• \( \theta \): the angle measured from the positive x-axis.

This system is particularly useful for graphing curves that have rotational symmetry or involve angles naturally, like conic sections.
In the given exercise, the equation \( r = \frac{6}{2 + \sin(\theta + \pi / 6)} \) is a representation in polar form. The function involves \( \theta + \pi/6 \), which suggests that the conic is rotated by \( \pi/6 \) radians. By understanding polar coordinates, we simplify complex relationships between points and angles, allowing us to explore curves in new, insightful ways.

Graphing Utilities

Graphing utilities are tools, often software or online applications, that allow for visual representation of mathematical functions and equations. These tools are invaluable for students and professionals alike as they:

• Simplify the plotting of complex equations.
• Provide immediate visual feedback to tweak and understand functions better.
• Enable exploration of mathematical structures across various coordinate systems, like polar.

For the exercise, inputting the equation \( r = \frac{6}{2 + \sin(\theta + \pi / 6)} \) into a graphing utility set to polar mode will show the curve's behavior clearly. This visual aid helps check your work, identify the conic section, and see the rotational shift. Such tools enhance comprehension by depicting what might be hard to visualize just through pen and paper.

Trigonomic Transformations

Trigonometric transformations involve changes in trigonometric functions to reveal hidden characteristics or simplify the equations. In the context of this exercise, a transformation is noted in the sine function: \( \sin(\theta + \pi/6) \). This indicates a rotation.Key Characteristics:

• Phase Shift: The expression \( \theta + \pi/6 \) represents a horizontal shift, affecting where the sine wave begins.
• Behavior Change: These transformations adjust the graph's shape or orientation.
• Application: They are crucial for identifying rotations in polar coordinates.

By adjusting \( \theta \) with \( \pi/6 \), the function describes how the conic section rotates on the polar plane. Understanding this allows control over designing or interpreting graphs in mathematics or physics. Trigonometric transformations help uncover deeper insights into equation behaviors, like periodicity and symmetry across different domains.