Question

Graph the polar function on the given interval. \(r=\cos (\theta / 3), \quad[0,3 \pi]\)

Short answer

Graphing \(r = \cos(\theta / 3)\) from \([0, 3\pi]\) creates a multi-looped spiral.

Step-by-step solution

Understanding the Function

The function given is in polar form: \(r = \cos(\theta / 3)\). Here, \(r\) is the radial coordinate, and \(\theta\) is the angular coordinate. We need to plot the part of the cosine function from \(\theta = 0\) to \(\theta = 3\pi\).

Analyze the Range of \(\theta\)

Convert the interval \([0, 3\pi]\) for \(\theta\) to \([0, \pi]\) in terms of the cosine argument by noting that \(\theta/3\) covers one full period and a half of the cosine curve. This means analyzing from \(0\) to \(\pi\), including peaks and troughs.

Key Points on the Cosine Curve

Find key points within the interval: at \(\theta = 0\), \(\theta = \pi\), and \(\theta = 2\pi\), the value of \(\cos(\theta/3)\) will be 1, 0, and -1 respectively. Repeat up to \(\theta = 3\pi\), which corresponds to \(\pi\) in \(\theta/3\).

Plotting the Graph

Start plotting from \(\theta = 0\) to \(3\pi\), marking the points derived: at \(\theta = 0\), \(r = 1\); at \(\theta = \pi\), \(r = 0\); at \(\theta = 2\pi\), \(r = -1\); and so on. Sketch the polar plot by connecting these points smoothly as a cosine curve in polar coordinates would behave, repeating the cycle for \(\theta/3\) as required.

Finalizing the Graph

Given that \(\theta\) from 0 to \(3\pi\) results in one and a half cycles of \(\cos(\theta/3)\), finalize the graph ensuring the symmetry of the cosine function is maintained in spiral arms.

Key concepts

Cosine Function

The cosine function is a fundamental trigonometric function that represents the horizontal coordinate on the unit circle. It cycles between -1 and 1 over its total period, making it periodic with a well-defined pattern. In polar coordinates, the cosine function can dictate the distance from the origin—this distance is referred to as the radial coordinate.

• Amplitude: It indicates the maximum distance the wave reaches from its mean position, which in the case of \( = \cos(\theta/3)\) is 1.
• Period: Normally for \( = \cos(\theta)\), the period is \(2\pi\), however, due to the \(\theta/3\) factor, it is \(6\pi\), indicating it takes longer to complete a cycle.
• Behavior: It starts at its peak when \(\theta = 0\), descends to zero, reaches the negative peak, ascends back to zero, and goes back to the peak in one full cycle.

Understanding the cosine function is key to grappling with polar graphing, helping to anticipate how the plot will unfold in these types of equations.

Graphing Polar Functions

When graphing a polar function like \(r = \cos(\theta/3)\), the approach is slightly different from Cartesian plotting. Polar functions rely on the distance from the origin and the angle made with the positive x-axis.

• Radial Line Motion: As \(\theta\) increases, \(r\) dictates how far from the origin the point is.
• Symmetry and Shape: The cosine's looping and peaking nature often leads to knotty, symmetric shapes, particularly when \(r\) can be negative, allowing tracing in the opposite direction.
• Angles and Spirals: Since each angle corresponds to a position in the radial movement, it affects the resulting spiral curve prominently.

Accurate plotting of key points such as the peaks, midpoints, and zeroes is necessary to sketch the complete shape accurately. These points are crucial for guiding the eventual shape of the graph in polar form.

Radial Coordinate

The radial coordinate \(r\) in polar coordinates represents the directed distance from the origin to the point of interest. In functions like \(r = \cos(\theta/3)\), \(r\) can be negative, indicating the point is in the opposite direction along the same line.

• At \(\theta = 0\): Points lie at a maximum distance, equivalent to the amplitude of the cosine function.
• At \(\theta = \pi/2\): The point matches the sine wave's maximum upward stretch indicating zero distance along the cosine wave.
• Negative Values: When the cosine yields a negative \(r\), it signifies tracing a point oppositely. This alters the graph's comprehension from mere positive spirals to encompassing backward loops.

Understanding how \(r\) functions in polar coordinates expands the ability to manipulate and graph polar functions correctly, creating accurate, descriptive graphs.

Polar Graphs

Polar graphs are a visually dynamic form, distinguished by their circular characteristics, contrasting the rectilinear nature of Cartesian graphs. They rely heavily on angles and radii.

• Curves and Loops: Interactions of \(\theta\) and \(r\) lead to rounded, looping shapes characteristic of polar graphs.
• Unique Symmetry: Many polar graphs maintain a certain symmetry, allowing them to mirror across a central line—such as the line \(\theta = 0\)—depending on the periodic function used.
• Completing Patterns: To complete the graphical pattern, recognizing repeating patterns in the function within a given interval, such as \(0\) to \(3\pi\), shows these symmetries fully.

Mastering polar graphs involves anticipating how radial adjustments and the cycling of angles impact plot shape. This understanding brings clarity and precision to graphing, particularly with functions like cosine, enhancing analytical skills in exploring complex, looping polar paths.