Problem 87 - Algebra and Trigonometry

Question

Graph the spiral \(r=\theta .\) Use a \([-48,48,6]\) by \([-30,30,6]\) viewing rectangle. Let \(\theta\) min \(=0\) and \(\theta \max =2 \pi,\) then \(\theta \min =0\) and \(\theta \max =4 \pi,\) and finally \(\theta \min =0\) and \(\theta \max =8 \pi\)

Step-by-Step Solution

Answer

Different segments of the spiral \(r=\theta\) were graphed for different \(\theta\) ranges. The spiral made one complete turn for \(\theta\) max = \(2 \pi\), two turns for \(\theta\) max = \(4 \pi\), and four turns for \(\theta\) max = \(8 \pi\).

Step 1: Understand Polar Coordinates

In a polar coordinate system, the location of a point is determined by its distance from the origin (r) and the angle it forms with the positive x-axis (\(\theta\)). Here we have a spiral defined by \(r=\theta\). This means the distance from the origin increases proportionally as the angle from the positive x-axis increases.

Step 2: Graphing for \(\theta\) in range [0, \(2\pi\)]

Start by plotting the initial viewing rectangle [-48,48,6] by [-30,30,6], This means the x and y axis will range from -48 to 48 and -30 to 30 with tick marks every 6 units respectively. Then, plot the spiral \(r=\theta\) for \(\theta\) min = 0 and \(\theta\) max = \(2 \pi\). You'll see part of a spiral starting from the origin and making one complete rotation around.

Step 3: Graphing for \(\theta\) in range [0, \(4\pi\)]

Next, we’ll extend the range of \(\theta\) to maximum \(4 \pi\). Plot the graph with same viewing rectangle as in step 2. You will notice the spiral making two complete turns this time, as \(\theta\) range has extended.

Step 4: Graphing for \(\theta\) in range [0, \(8\pi\)]

Finally, the range of \(\theta\) extends to maximum \(8 \pi\). Plot the graph with same viewing rectangle as before. Now you'll observe four complete turns of the spiral.

Key Concepts

Graphing Spirals

Graphing spirals in a polar coordinate system can be a fascinating task because of the unique shapes they create. In this exercise, we are asked to graph the spiral described by the equation \(r = \theta\). The value of \(r\) represents the distance from the origin, and \(\theta\) indicates the angle with respect to the positive x-axis. As \(\theta\) increases, \(r\) also increases, resulting in a spiral.

To graph this, we begin with an initial angle \(\theta = 0\) and increment it up to a certain maximum value. For \(r = \theta\), it forms an Archimedean spiral where each loop of the spiral is evenly spaced from the center. Each increment in \(\theta\) results in a uniform increase in \(r\), causing the spiral to wind further outward.

To visualize these spirals:

• For \(\theta\) ranging from 0 to \(2\pi\), the spiral makes one complete rotation around the origin.
• For \(\theta\) from 0 to \(4\pi\), the spiral makes two full rotations, illustrating an extended version of the one-rotation spiral.
• Finally, for \(\theta\) from 0 to \(8\pi\), it completes four rotations, showcasing the expanding nature of spirals.

Observing the spiral evolve as \(\theta\) increases can help you understand how polar graphs differ from traditional Cartesian graphs.

Angle Measurement

Understanding angle measurement is crucial when dealing with polar coordinates and spirals. The angle \(\theta\) is measured in radians, which is a unit of angular measurement. One complete revolution around a circle is \(2\pi\) radians.

When we describe the spiral through varying angle ranges, it’s essential to know what these ranges mean in the context of a circle:

• \(\theta = 2\pi\) corresponds to 360 degrees, indicating one full rotation.
• \(\theta = 4\pi\) is equivalent to two full rotations, or 720 degrees.
• \(\theta = 8\pi\) equates to four full rotations or 1440 degrees.

Knowing these conversions helps in plotting and understanding the extent of the spiral rotations when graphing them in the polar coordinate system. The ability to think in radians rather than degrees often simplifies calculations related to circles and mathematics involving trigonometry.

Viewing Rectangles

A viewing rectangle is a specified range for plotting graphs, acting like a window through which you visualize the curve. For the given spiral, the viewing rectangle is set as \([-48, 48, 6]\) by \([-30, 30, 6]\).

This means:

• The horizontal (x-axis) values range from -48 to 48, and vertical (y-axis) values range from -30 to 30.
• Tick marks or grid lines are placed at intervals of 6 units on both axes, making it easier to locate specific points.

Setting appropriate viewing rectangles is essential to clearly observe complex graphs such as spirals. If the rectangle is too small, you might miss parts of the graph. Conversely, if it’s too large, the graph may become unclear.

A well-chosen viewing rectangle provides better context and clarity, allowing you to see the entire spiral’s rotations and expansions within the given \(\theta\) limits. As you adjust \(\theta\) max values in the exercise, the spiral unfolds in the defined window, showcasing different segments and patterns.