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Chapter 6: Problem 14

Sketch the graph of each polar equation. $$r=5 \cos 3 \theta$$

Short Answer

Expert verified
To sketch the graph of the function, generate a table of values of r for different 胃 from 0 to \(2 \pi\) and then plot these coordinates. The resulting graph would be a rose curve with 3 petals.

Step by step solution

01

Set up a table

To start off, consider the trigonometric function and set up an interval from 0 to \(2 \pi\) since \(\cos\) has a period of \(2 \pi\). Create a table with 胃 in one column and r in the next. Divide the interval into small sub-intervals, say \(\frac{ \pi }{ 6 }\), and write down their corresponding r values according to the equation \(r=5 \cos 3 \theta\).
02

Generate Polar Coordinates

Now, convert the table to polar coordinates. The angles 胃 can remain the same, but multiply the r values by the result of \(\cos 3 \theta\) in order to match the form of the polar coordinates \( ( r , \theta ) \) given by the equation.
03

Plotting the graph

Use the polar coordinates from the results in step 2 to plot the graph. Note if the value of r is negative, the direction is opposite. Plot all the points generated and then connect the points with a smooth curve to form the graph. It may be useful to start from 胃 = 0 and proceed to plot points in an anti-clockwise direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Graph
A polar graph represents a function in a circular coordinate system, rather than the traditional Cartesian coordinate system. In a polar graph, points are determined by a distance from a reference point and an angle from a reference direction. This allows for the representation of curves that are difficult to express in rectangular coordinates.

  • Origin as the Pole: The pole is equivalent to the origin in Cartesian coordinates, serving as the center point of our circle.
  • Plotting based on Angles: Instead of using x and y coordinates, polar graphs use an angle \(\theta\) and a radius r.
  • Interpreting R and Theta: The radius, r, determines how far you move from the pole, while \(\theta\) tells you the direction.
This graphing technique allows for picturesque representations of non-linear equations, making them incredibly useful for visualizing trigonometric functions.
Trigonometric Function
Trigonometric functions are functions related to the angles and sides of right-angled triangles. They are essential in describing periodic phenomena and appear in many areas of mathematics and physics.

  • Basic Functions: The primary trigonometric functions are sine, cosine, and tangent. Each one has its particular characteristics and applications.
  • Periodicity: Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For cosine, this period is \(2 \pi\).
  • Use in Polar Equations: Functions like \(\cos\) and \(\sin\) are widely used in polar equations because they easily describe circular shapes.
Trigonometric functions are powerful tools for modelling anything that exhibits cyclical behavior, making them foundational for polar graphing and complex analysis.
Cosine Function
The cosine function is a core trigonometric function with unique properties that make it particularly suitable for polar graphing.

  • Definition: The cosine of an angle in a right-angle triangle is the length of the adjacent side divided by the hypotenuse. It also correlates to the x-coordinate on the unit circle.
  • Symmetry and Periodicity: Cosine is an even function with a period of \(2 \pi\). This means \(\cos(\theta) = \cos(-\theta)\), and it repeats its pattern every \(2 \pi\).
  • Graph Characteristics: In polar coordinates, equations involving cosine like \(r = 5 \cos 3 \theta\) exhibit symmetrical and often flower-like shapes called "rose curves" which arise due to the nature of the cosine function and its periodicity.
Understanding the characteristics of the cosine function helps in predicting and sketching the path the curve takes in a polar graph, offering a visual and mathematical symmetry.

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