Polar Coordinates and Equations

Symmetric Tests of a Polar Equation

1. Replace θ by -θ in a polar equation. If the equation is equivalent, the graph of the polar equation is symmetric with respect to the polar axis (x-axis) 2. Replace θ by π-θ. If the equation is equivalent, the graph is symmetric with respect to θ = π/2 (y-axis) 3. Replace r by -r. If the equation is equivalent, the graph is symmetric with respect to the pole (origin)

Polar Axis

A fixed ray with origin at the pole, usually horizontal, from which an angle can be established. Analogous to the x-axis in a Cartesian coordinate system.

Corresponding Polar Coordinates

A point with polar coorinates (r, θ), θ in radians, can also be represented by either: (r, θ + 2πk) or (-r, θ + π + 2πk) Where k is any integer.

Limaçon (inner loop)

A polar equation of the form r = a + b cos θ r = a + b sin θ r = a - b cos θ r = a - b sin θ Where a > 0, b > 0 and a < b. The graph is shaped similarly to a heart and passes through the pole (origin) twice, which forms the inner loop.

Limaçon (no inner loop)

A polar equation of the form r = a + b cos θ r = a + b sin θ r = a - b cos θ r = a - b sin θ Where a > 0, b > 0 and a > b. The graph is shaped similarly to a heart but does not pass through the pole (origin).

Rose Curve

A polar equation of the form r = a cos (nθ), r = a sin (nθ), a ≠ 0 and whose graphs are rose-shaped. If n ≠ 0 is even, the rose has 2n petals; if n ≠ ±1 is odd, the rose has n petals.

Logarithmic Spiral

A polar equation of the form r = e^(θ/5) → θ = 5 ln r whose graphs spiral infinitely both toward and away from the pole.

Lemniscate

A polar equation of the form r² = a² sin (2θ) r² = a² cos (2θ) where a ≠ 0 and whose graphs are propeller shaped.

Cardioid

A polar equation of the form r = a(1 + cos θ) r = a(1 + sin θ) r = a(1 - cos θ) r = a(1 - sin θ) where a > 0. The graph is shaped similarly to a heart and passes through the pole (origin).

Graph of a Polar Equation

All points whose polar coordinates satisfy a particular polar equation.

Polar Equation

An equation whose variables are polar coordinates.

Conversion from Polar to Rectangular Coordinates

If P is a point with polar coordinates (r,θ), the rectangular coordinates (x,y) of P are given by x = r cos θ y = r sin θ

Conversion from Rectangular to Polar Coordinates

If P is a point with rectangular coordinates (x,y): 1. Find the quadrant P lies in. 2. If x = 0 or y = 0, r can be found by graphing the point; otherwise, r = √(x² + y²) 3. If x = 0 or y = 0, θ can be found by graphing the point; otherwise, a. If P lies in quadrant I or IV: θ = tan⁻¹(y/x) b. If P lies in quadrant II or III: θ = π + tan⁻¹(y/x)

Straight Line Graphs of a Polar Equation

Let a be a nonzero real number. The graph of equation r sin θ = a Is a horizontal line θ units above the pole if a ≥ 0 and |a| units below the pole if a < 0. The graph of equation r cos θ = a Is a vertical line units to the right of the pole if a ≥ 0 and |a| units to the left of the pole if a < 0.

Circle Graphs of a Polar Equation

Let a be a positive real number. Then (a) r = 2a sin θ Circle: radius a, center at (0,a) in rectangular coordinates (b) r = -2a sin θ Circle: radius a, center at (0,-a) in rectangular coordinates. (c) r = 2a cos θ Circle: radius a, center at (a,0) in rectangular coordinates. (d) r = -2a cos θ CIrcle: radius a, center at (-a,0) in rectangular coordinates.

Polar Coordinate

Ordered pair of numbers (r, θ) which represents a point in the polar coordinate system, where if r is the distance from the point to the pole and θ is the angle formed between the polar axis and a ray extending from the pole through the point.

Pole

The point in a polar coordinate system used as a reference. Analogous to the origin in a Cartesian coordinate system.

Transforming Equations between Polar and Rectangular Forms

Use formulas r² = x² + y² tan θ = y/x x = r cos θ y = r sin θ To convert from polar to rectangular and vise versa. Multiplying both sides of the equation by r to get r² and squaring both sides of the equation are common techniques to transform an equation from one to the other.