Graphing Polar Equations: Circles, Limacons, Roses, and Lemniscates

10.2 Graphing Polar Equations

Sketching Equations of the Form r = a cos θ, r = a sin θ, θ = a, or r = a

Polar equations can represent lines and circles in the polar coordinate system. Understanding their forms helps in sketching their graphs efficiently.

• Vertical Lines: The graph of r cos θ = a is a vertical line.

• Horizontal Lines: The graph of r sin θ = a is a horizontal line.

• Lines with Slope: The graph of r cos θ + r sin θ = a is a line with slope $m = -\frac{A}{B}$ and y-intercept $b = \frac{a}{B}$.

• Lines through the Pole: The graph of θ = a is a line through the pole making an angle a with the polar axis.

Sketching Equations of the Form r = a, r = a sin θ, r = a cos θ

These equations represent circles in the polar coordinate system, with their centers and radii determined by the coefficients.

• Circle Centered at the Pole: r = a is a circle centered at the pole with radius $|a|$.

• Circle Centered on the Polar Axis: r = a cos θ is a circle centered $\frac{a}{2}$ units from the pole on the line θ = 0 with radius $\frac{|a|}{2}$.

• Circle Centered on the Line θ = π/2: r = a sin θ is a circle centered $\frac{a}{2}$ units from the pole on the line θ = \frac{\pi}{2} with radius $\frac{|a|}{2}$.

Sketching Equations of the Form r = a ± b sin θ and r = a ± b cos θ

These equations produce limacons, which are a family of polar curves with varying shapes depending on the ratio $\frac{a}{b}$.

• Cardioid: If $\frac{a}{b} = 1$, the graph is a cardioid.

• Limacon with Inner Loop: If $0 < \frac{a}{b} < 1$, the graph is a limacon with an inner loop.

• Limacon with Dimple: If $1 < \frac{a}{b} < 2$, the graph is a limacon with a dimple.

• Limacon with No Inner Loop and No Dimple: If $\frac{a}{b} \geq 2$, the graph is a limacon with no inner loop and no dimple.

Example: The graph of r = 2 + 2 sin θ is a cardioid, while r = 1 + 2 cos θ is a limacon with an inner loop.

Steps for Sketching Limacons (r = a ± b sin θ and r = a ± b cos θ)

1. Identify the general shape using the ratio $\frac{a}{b}$.

2. Determine the symmetry:

   • If the equation is of the form r = a ± b sin θ, the graph is symmetric about the line θ = \frac{\pi}{2}.

   • If the equation is of the form r = a ± b cos θ, the graph is symmetric about the polar axis.

3. Plot points corresponding to the quadrant angles and use symmetry to complete the graph.

Sketching Equations of the Form r = a sin(nθ) and r = a cos(nθ)

These equations produce rose curves, which are polar graphs with petal-like structures. The number and orientation of petals depend on the value of n.

• Rose with Odd n: The graph of r = a sin(nθ) or r = a cos(nθ) with odd n has n petals.

• Rose with Even n: The graph with even n has 2n petals.

• Petal Length: Each petal has length $|a|$.

Example: r = 2 sin(3θ) produces a rose with 3 petals, while r = 2 cos(4θ) produces a rose with 8 petals.

Steps for Sketching Rose Curves (r = a sin(nθ) and r = a cos(nθ))

1. Identify the number of petals:

   • If n is even, there are 2n petals.

   • If n is odd, there are n petals.

2. Determine the length of each petal: $|a|$.

3. Determine all angles where an endpoint of a petal lies.

4. Substitute each angle into the equation to find the corresponding radius.

5. Determine symmetry:

   • If the equation is of the form r = a sin(nθ), the graph is symmetric about the line θ = \frac{\pi}{2}.

   • If the equation is of the form r = a cos(nθ), the graph is symmetric about the polar axis.

6. Draw each petal to complete the graph.

Sketching Equations of the Form r² = a² sin(2θ) and r² = a² cos(2θ)

These equations produce lemniscates, which are figure-eight shaped curves symmetric about the pole.

• Lemniscate Symmetric about the Pole and Line θ = π/4: r² = a² sin(2θ)

• Lemniscate Symmetric about the Polar Axis and Vertical Line: r² = a² cos(2θ)

• Length of Loops: Each loop has length $|a|$.

Example: r² = 4 sin(2θ) and r² = 4 cos(2θ) both produce lemniscates with loops of length 2.

Summary Table: Types of Polar Graphs

Equation Form	Graph Type	Key Features
r = a	Circle	Centered at pole, radius $|a|$
r = a cos θ, r = a sin θ	Circle	Centered on axis, radius $\frac{|a|}{2}$
r = a ± b sin θ, r = a ± b cos θ	Limacon/Cardioid	Shape depends on $\frac{a}{b}$
r = a sin(nθ), r = a cos(nθ)	Rose	n or 2n petals, length $|a|$
r² = a² sin(2θ), r² = a² cos(2θ)	Lemniscate	Figure-eight, loop length $|a|$

Additional info: These notes cover the main types of polar graphs encountered in college trigonometry, including their equations, shapes, and sketching strategies. For exam preparation, practice sketching each type and identifying symmetry and key features.