Graphing Polar Equations

Types of Polar Equations

Circles
- Equation: r = a cos(θ) or r = a sin(θ)
- Cosine: Centered on x-axis
  - Positive a: Circle on the right
  - Negative a: Circle on the left
- Sine: Centered on y-axis
  - Positive a: Circle above x-axis
  - Negative a: Circle below x-axis
- Diameter = a, Radius = a/2
- Example: r = 4 cos(θ) forms a circle with diameter 4
- Area of circle: πr² where r = a/2
Limaçons
- Equation: r = a ± b cos(θ) or r = a ± b sin(θ)
- Shape depends on a/b ratio:
  - Inner Loop: a/b < 1
  - Cardioid: a/b = 1
  - Dimpled Limaçon: 1 < a/b < 2
  - No Dimple: a/b ≥ 2
- Direction:
  - Positive cos(θ): Opens right
  - Negative cos(θ): Opens left
  - Positive sin(θ): Opens up
  - Negative sin(θ): Opens down
- Important points: x-intercepts (a±b), y-intercepts (±a)
Rose Curves
- Equation: r = a sin(nθ) or r = a cos(nθ)
- Number of Petals:
  - Even n: 2n petals
  - Odd n: n petals
- Sine: Petals often not on x-axis
- Cosine: Petals on x and y axes when n is even
- Example: r = 2 sin(3θ) creates a rose with 3 petals
Lemniscates
- Equations:
  - r² = a² cos(2θ)
  - r² = a² sin(2θ)
- Cosine symmetry: Pole, polar axis, and line θ = π/2
- Sine symmetry: Symmetric about the pole

Polar Coordinate System: r represents radial distance, θ angle from the polar axis
Graphing Tips:
- Identify equation type
- Determine symmetry and direction
- Calculate specific points (intercepts, petals)
- Consider using auxiliary circles to determine petal lengths or centers

Useful in plotting complex periodic behaviors in physics, engineering, and mathematics
Visual aids for understanding trigonometric transformations in polar coordinates

Remember that polar graphs can flip or change orientation based on the sign of a or the function (cosine vs. sine).
Understanding the behavior of polar equations helps in visualizing and interpreting complex systems.