Polar Coordinates: Described as (R, θ) where R is the magnitude from the origin (pole) and θ is the direction in radians counterclockwise from the positive x-axis.
Rectangular Coordinates: Regular XY format, like (4, 8) means move right 4 and up 8.
Converting Between Coordinates
Rectangular to Polar:
Magnitude (R) = sqrt(X^2 + Y^2)
Angle (θ) = tan⁻¹(Y/X)
Polar to Rectangular:
X = Rcos(θ)
Y = Rsin(θ)
Complex Numbers and Polar Coordinates
Complex Number (Rectangular Form): a + bi
Polar Form of Complex Number: Rcos(θ) + iRsin(θ)
Conversion to Polar Form
Given a complex number a + bi,
Magnitude (R) = sqrt(a^2 + b^2)
θ = tan⁻¹(b/a)
Graphing Polar Functions
Circle: R = acos(θ) or R = asin(θ)
Open upwards or downwards for sine, left or right for cosine.
Magnitude (a) determines the size.
Rose: R = acos(nθ) or R = asin(nθ)
n even: 2n petals
n odd: n petals
Location starts as sine or cosine dictates (starts axis-aligned for cosine).
Lemniscate (Infinity Symbol): Presented when the equation format is R = a ± b*cos(θ) or sin(θ)
Type and orientation of lemniscate depend on the constants a and b.
Identifying Polar Graph Features
The magnitude (a) and angle (θ) parameters influence the polar graph's shape significantly:
Circle magnitudes stretch or compress the circle.
In rose and lemniscate graphs, the parameter n dictates the number of primary features (petals or loops).
Additional parameters can cause vertical shifts, affecting the graph's origin-centered symmetry.
Tracing and Plotting on Polar Graphs
A graph can be traced or plotted by examining specific intervals of θ and noting how R changes:
Positive R values plot directly in the angle's direction.
Negative R values plot in the opposite direction, akin to reflecting over the pole.
Converting and Evaluating Polar Functions
To evaluate or convert polar functions, especially for finding specific polar points (R, θ) or transitioning between polar and rectangular forms, apply the basic trigonometric identities and properties pertinent to polar coordinates.