Question

Sketch a graph of the polar equation and identify any symmetry. $$r=2\cos (3\theta )$$

Short answer

The graph of $r=2\cos (3\theta )$ has 6 petals and exhibits symmetry about the x-axis, y-axis, and $\theta =\frac{\pi }{3}$.

Step-by-step solution

Understand Polar Coordinates

Polar coordinates use $r$ to represent the distance from the origin and $\theta$ as the angle from the positive x-axis. In the equation $r=2\cos (3\theta )$, $r$ is defined in terms of $\theta$. To understand what this graph might look like, we need to consider values of $\theta$ and compute the corresponding $r$.

Recognize Key Features

Identify key features of the polar equation $r=2\cos (3\theta )$. The equation has the form $r=a\cos (n\theta )$, where $a=2$ and $n=3$. This indicates a rose curve with $2n=6$ petals, suggesting that there will be 6 symmetrical loops around the origin.

Test Points at Different Angles

Calculate $r$ for specific $\theta$ values to help sketch the graph. For example, at $\theta =0$, $r=2$ (cosine is 1); at $\theta =\frac{\pi }{6}$, $r=2\cos (\frac{\pi }{2})=0$; at $\theta =\frac{\pi }{3}$, $r=-2$ (cosine is -1). Continue testing values at regular intervals, especially multiples of $\frac{\pi }{6}$, because they are easier to plot and reveal the symmetry.

Plot Points and Sketch the Graph

Plot the calculated points on polar graph paper. Remember that negative $r$ values indicate the point lies in the direction opposite $\theta$. Use the points to sketch a smooth curve through the points, forming a rose curve with 6 petals. Observe the symmetry: the graph is symmetric with respect to the origin.

Identify Symmetry

The symmetry in polar equations like $r=2\cos (3\theta )$ can be checked by substituting $\theta$ with negative angles or by looking for reflections about the x-axis (horizontal axis). As the graph has 6 identical petals equally spaced, it shows symmetry about the x-axis, the y-axis, and the line $\theta =\frac{\pi }{3}$.

Key concepts

Polar Equation

A polar equation expresses a relationship between the variable radius, $r$, and the angle, $\theta$. In the case of our polar equation $r=2\cos (3\theta )$, the radius $r$ depends on the cosine of three times the angle $\theta$. Polar coordinates differ from Cartesian coordinates because they pinpoint a location based on its distance from a central point and the direction, or angle, from a fixed axis.

Polar equations often take forms that describe elegant curves such as circles, spirals, or more complex shapes like rose curves. In this system, different functions can create strikingly different patterns due to the trigonometric nature of the equations. This is why understanding how to interpret and graph polar equations can be both challenging and rewarding.

Symmetry in Polar Graphs

Understanding symmetry can simplify sketching polar graphs and analyzing their properties. A polar graph shows symmetry if it is a mirror image of itself about a certain line or point. There are typically three types of symmetry to consider:

• Symmetry about the x-axis: If replacing $\theta$ with $-\theta$ in the equation produces the same form, the graph is symmetric about the x-axis.
• Symmetry about the y-axis: This symmetry occurs if changing the sign of $r$ yields the same equation.
• Symmetry about the origin: If changing $\theta$ to $\theta +\pi$ gives the same equation, the graph is symmetric about the origin.

For the equation $r=2\cos (3\theta )$, our observation reveals symmetry about both axes and the origin, owing to its structure and trigonometric nature.

Rose Curve

A rose curve is a specific type of polar graph that resembles the petals of a rose. The general equation for a rose curve can be expressed as $r=a\cos (n\theta )$ or $r=a\sin (n\theta )$, where $n$ dictates the number of petals and $a$ influences the size.

For our equation $r=2\cos (3\theta )$, the number of petals is determined by the value $2n$. Thus, we expect 6 petals (since $n=3$, resulting in $2\times 3=6$ petals). If $n$ were odd, the equation would have $n$ petals. The repetition in the cosine function helps create this visually pleasing pattern, engaging in a kind of symmetry that is simple to recognize.

Plotting Polar Graphs

When plotting polar graphs, it is crucial to correctly determine the points from the polar equation. Here’s a simple guide to plotting polar graphs:

• Select Values: Choose several values for $\theta$ and compute their corresponding $r$ values. Selecting these points strategically is key.
• Consider Symmetry: Use the symmetry properties to predict and simplify plotting, reflecting the points around the axes as needed.
• Sketch the Curve: Connect the plotted points smoothly, ensuring the features of the curve, like petals or loops, are clear.

For the polar equation $r=2\cos (3\theta )$, plot critical angles such as $\theta =0,\frac{\pi }{6},\frac{\pi }{3}$, and others around the same interval. Plotting in polar graphs can initially be challenging, but using symmetry and calculated points ensures accuracy and elegance. Don’t forget that negative values of $r$ imply points in the opposite direction of $\theta$.