Problem 40 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=2 \cos 2 \theta$$

Step-by-Step Solution

Answer

It is a circle with radius 2 and its center at the origin, symmetrical about the x-axis.

Step 1: Finding Zeroes

The zeroes are achieved when r = 0. From the equation \(r= 2 \cos 2 \theta\), the zero occurs when \(\cos 2 \theta = 0\). By applying the trigonometric properties, calculate the values for \(\theta\) which causes r to be zero. This can be computed as \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4} , \frac{7\pi}{4}\).

Step 2: Symmetry

To determine the symmetry of a polar function, check if the functions are equal where you substitute \(-\theta\) for \(\theta\). If they are equal, then it is symmetrical with regards to the x-axis. In this case, \(r= 2 \cos 2(-\theta) = 2 \cos 2\theta\), which is indeed equal to the original function, therefore our graph is symmetrical with regards to the x-axis.

Step 3: Maximum r-values

The maximum r-value is the maximum value of \(2\cos2\theta\). The maximum value of a cosine function is 1. Therefore, the maximum r-value for this function is 2*1=2.

Step 4: Sketching the graph

Plot the zeroes, maximum points, and add additional points as needed while maintaining symmetry. Be sure to draw a smooth curve and not sharp corners or points where the graph is not defined.

Key Concepts

Trigonometric Functions

Trigonometric functions are at the heart of understanding polar equations, like the one given in the exercise \(r = 2 \cos 2\theta\). Let's consider cosine, one of the basic trigonometric functions, which relates an angle to the length of the adjacent side over the length of the hypotenuse in a right-angled triangle. In the context of polar graphs, the cosine function helps in determining the radial distance \(r\) from the origin to a point on the curve at a given angle \(\theta\).

Remember that the cosine function oscillates between -1 and 1, and due to its periodic nature, it repeats its values every \(2\pi\) radians. This repetition is key when graphing as it dictates the pattern that will emerge on the polar plot. To better understand the graph of the given polar equation, it helps to first visualize the graph of \(y=\cos x\) on a Cartesian coordinate system, observing how the cosine curve rises and falls as it moves along the \(x\)-axis.

Graph Symmetry

Graph symmetry simplifies the process of graphing complex equations by revealing inherent patterns. In polar coordinates, there are several types of symmetry to consider: symmetry with respect to the line \(\theta=0\) (x-axis), symmetry with respect to the pole (origin), and symmetry with respect to the line \(\theta=\frac{\pi}{2}\) (y-axis).

In the exercise, we check for symmetry by substituting \(\theta\) with \( -\theta \). If the equation remains unchanged, as it does with \(r=2\cos 2\theta\), it indicates that the graph is symmetric about the polar axis (x-axis). This knowledge allows for half the effort when sketching — what we plot for positive \(\theta\) also applies to negative \(\theta\), reflecting across the polar axis. Equipped with this insight, creating a polar graph becomes significantly more manageable.

Polar Coordinates

Polar coordinates are another method for expressing the position of a point, using a radius and an angle, rather than x and y coordinates as in Cartesian coordinates. In polar coordinates, each point on the plane is defined by \( (r, \theta) \) where \( r \) is the radial distance from the origin (pole) and \( \theta \) is the angle from the polar axis (positive x-axis).

The beauty of polar coordinates lies in their ability to naturally describe circular and spiral shapes, making them especially useful for problems involving rotations and angles. Unlike Cartesian coordinates that give a rectangular grid, a polar grid consists of circles centered at the pole and lines emanating outward at various angles. In practice, to plot a point, you first move radially outward from the origin a distance of \( r \) and then rotate by the angle \( \theta \) relative to the polar axis.

Maximum r-values

The maximum r-values in a polar equation represent the largest distance from the origin (pole) achieved by the function. In the given exercise, our function \( r = 2 \cos 2\theta \) depends on \( \cos 2\theta \), which has a highest possible value of 1. Therefore, the maximum r-value for \( r \) is obtained by multiplying the maximum of the cosine function by the coefficient in front of it — here, it is 2. The maximum r-value is thus 2.

Knowing the maximum r-value is pivotal when graphing a polar equation as it defines the outer boundary of the graph. It's also used to determine the scale of the radial axis in the polar graph. When sketching the graph, it's essential to highlight these maximum points since they represent key features in the overall shape of the graph. They serve as a guide to accurately portray the function's behavior as \( \theta \) varies.