Graphing and Converting Polar Equations in Trigonometry

Polar Coordinates and Graphs

Introduction to Polar Coordinates

Polar coordinates provide an alternative way to describe the location of points in the plane using a radius and angle, rather than the traditional Cartesian (x, y) coordinates. This system is especially useful for representing curves and equations that have circular or rotational symmetry.

• Polar Coordinate: A point is represented as (r, θ), where r is the distance from the origin (pole) and θ is the angle measured from the positive x-axis (polar axis).

• Conversion: The relationship between polar and Cartesian coordinates is given by:

$x = r \cos \theta$ $y = r \sin \theta$

• To convert from Cartesian to polar:

$r = \sqrt{x^2 + y^2}$ $\theta = \arctan\left(\frac{y}{x}\right)$

Graphing Polar Equations

Example: r = 2 + 2cosθ

This equation represents a limaçon with an inner loop. To graph it, calculate r for various values of θ and plot the corresponding points.

• Step 1: Create a table of values for θ and r.

• Step 2: Plot each (r, θ) point on polar graph paper.

• Step 3: Connect the points smoothly to reveal the curve.

Example Table:

θ	r
0	4
π/2	2
π	0
3π/2	2
2π	4

Graph: The curve starts at (4, 0), passes through (2, π/2), reaches the pole at (0, π), and returns to (4, 2π).

Example: r = 3

This equation describes a circle centered at the pole (origin) with radius 3.

• For all values of θ, r remains constant at 3.

• Plotting all points (3, θ) for θ from 0 to 2π traces a circle.

Converting Between Polar and Cartesian Equations

Key Formulas

• From Polar to Cartesian:

$x = r \cos \theta$ $y = r \sin \theta$

• From Cartesian to Polar:

$r = \sqrt{x^2 + y^2}$ $\theta = \arctan\left(\frac{y}{x}\right)$

• Example: Convert the polar equation $r = 2 + 2\cos\theta$ to Cartesian form.

Multiply both sides by r:

$r = 2 + 2\cos\theta$ $r - 2\cos\theta = 2$ $r - 2\left(\frac{x}{r}\right) = 2$ $r^2 - 2x = 2r$ $r^2 - 2r - 2x = 0$ $(x^2 + y^2) - 2\sqrt{x^2 + y^2} - 2x = 0$

Additional info: This equation can be further manipulated to standard forms for conic sections if needed.

Summary Table: Common Polar Graphs

Equation	Graph Type	Description
r = a	Circle	Circle of radius a centered at the pole
r = a + b cosθ	Limaçon	Variety of shapes depending on a and b
r = a sin nθ	Rose Curve	n petals if n is odd, 2n if n is even
θ = α	Line	Straight line at angle α from the pole