Graph each polar equation. Also, identify the type of polar graph.
r² = 4 cos 2θ

Verified step by step guidance

1. Step 1: Recognize the form of the polar equation. The given equation is \( r^2 = 4 \cos 2\theta \). This is a type of polar equation that can represent a conic section, specifically a limaçon or a rose curve, depending on the transformation.
2. Step 2: Simplify the equation. Start by expressing \( r \) in terms of \( \theta \). Take the square root of both sides to get \( r = \pm \sqrt{4 \cos 2\theta} \). This indicates that the graph could have symmetry.
3. Step 3: Analyze the trigonometric function. The term \( \cos 2\theta \) suggests that the graph will have symmetry about the polar axis and will repeat every \( \pi \) radians due to the double angle.
4. Step 4: Identify the type of graph. The equation \( r^2 = 4 \cos 2\theta \) is characteristic of a lemniscate, which is a figure-eight shaped curve. This is because the equation is in the form \( r^2 = a^2 \cos 2\theta \), where \( a^2 = 4 \).
5. Step 5: Sketch the graph. To graph the lemniscate, plot points for various values of \( \theta \) and calculate the corresponding \( r \) values. Note that the graph will have two loops, one in each direction along the polar axis, due to the \( \cos 2\theta \) term.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Polar Equations

Polar equations express relationships between the radius 'r' and the angle 'θ'. The given equation, r² = 4 cos 2θ, is a type of polar equation that can represent various shapes, such as roses or lemniscates, depending on the coefficients and the trigonometric functions involved. Analyzing the equation helps determine the graph's characteristics.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of 'r' for different angles 'θ'. For the equation r² = 4 cos 2θ, one can derive the graph by calculating 'r' for various angles and observing the symmetry and periodicity of the function. Identifying the type of graph, such as a rose curve, requires understanding the behavior of the equation as 'θ' varies.