Polar Graphs, Coordinates,
Equations, and
Complex numbers!
A little bit of fun before we get back to the hard stuff! ( we need a break after that test
)
Polar Graphs:
Polar Graphs Exploration: You will use your calculator to make some pretty cool graphs and identify patterns.
Now back to reality…
Part 1: Polar Coordinates
Guided Notes
1. Graphing Polar Coordinates
2. Converting Polar Coordinates to Rectangular coordinates
3. Converting Rectangular coordinates to Polar coordinates
4. Distance between Polar coordinates
1. Graphing Polar Coordinates: ( r,  )
Rectangular Coordinates: (x,y) tell where a point is located by telling us how much
we move to the left/right (x-coordinate) and up/down (y-coordnate).
Polar coordinates tell us where a point is located by using a radius, r, and an angle, .
r = ________________________________________________________________________
 = ________________________________________________________________________
Examples:
Graph the following polar coordinates:
2. Converting Polar Coordinates, ( r,  ), to Rectangular Coordinates, ( x, y).
x = _____________________________
y = _____________________________
example: Convert from polar coordinates to rectangular coordinates.
3. Converting Rectangular Coordinates, (x, y) to Polar Coordinates, ( r,  ).
r = _____________________________
 = _____________________________
example: Convert from rectangular coordinates to polar coordinates.
4. Finding the distance between 2 polar coordinates.
Unfortunately there is not an easy formula to use: Steps 1. Convert to rectangular 2. Use distance form
Distance formula:________________________________________________
Practice:
Part 1: Polar Equations and Rectangular Equations
Guided Notes
1. Converting Rectangular Equations to Polar equations
2. Converting Polar Equations to Rectangular equations
1. Converting Rectangular Equations to Polar equations
All we use is simple substitution:
x = _______________________________________
y = _______________________________________
then simplify using trigonometry formulas if possible
Examples: Convert the following rectangular equations to polar equations
X + 5y = 8
x2 + (y + 3)2 = 9
2. Converting Polar Equations to Rectangular equations
Again, all we use is simple substitution, but we have 4 different substitution we can use:
_______________________________________
_______________________________________
_______________________________________
_______________________________________
then simplify if possible
Examples: Convert the following polar equations to rectangular equations.
r = - 6 cos 
 = π/4
Practice:
Part 3: Polar Coordinates and Complex numbers
Guided Notes
1. Converting Complex (rectangular) form to Polar form.
2. Converting Polar form to Complex (rectangular) form.
3. Using DeMoivre’s Theorem
1. Converting Complex (rectangular) form to Polar form.
Review: Plotting Complex numbers
i
3 -6i
r
Steps: 1. Plot the complex number: r2 = x2 + y2 and  = tan-1 (y/x)
2. Next write in the form z = r (cos  + i sin )
Example: Write the following complex numbers in polar form
2. Converting Polar form to Complex (rectangular) form
Steps: 1. Find the trig ratios of the polar form.
2. Distribute the “r”
Example: Write the following polar form into complex (rectangular) form
3. Using DeMoivre’s Theorem: Used to find powers of complex numbers (a + bi)n
DeMoivre’s Theorem: _________________________________________________________
Steps: 1. Re-write the complex number in polar form w/DeMoivre’s Theorem
2. Evaluate the trig ratios
3. Simplify and distribute
Examples: Use DeMoivre’s Theorem to find the power of the following complex numbers
Practice