Complex and Polars.Notebook November 18, 2015

Complex and Polars.notebook November 18, 2015

Complex Numbers in a Plane Graphing Complex Numbers: Complex Warm Up: Simplify the following complex numbers in standard form, a + bi, are graphed as an order pair (a, b) in the complex plane expressions. containing a real and imaginary axis. 1) 2) 3)

4) 5) 6) Absolute Value of a Complex Number: The absolute value of z = a + bi is | z | = | a + bi | =

Graph and find the absolute value of: Polar Coordinate Basics a) b) c) d) Rectangular Coordinates Polar Coordinates

x and y are horizontal and vertical lines, and their In the polar coordinate system, the origin is a point of intersection O is the origin. The location of fixed point O called the pole. The polar axis is a point P, (x, y), is where the x and y are the an initial ray from the pole, of a point P in a horizontal and vertical directed distances to the polar coordinate system can be identified by point. polar coordinates of the form (r, ), where r is the directed distance from the pole to the point and is the directed angle from the pole to the polar axis to OP.

Graphing Polar Graphing Polar Coordinates Coordinates • Positive ___ represents CCW • Positive ___ represents CCW • Negative ___ represents CW • Negative ___ represents CW • Positive r, P is on terminal side • Positive r, P is on terminal side • Negative r, P lies on the • Negative r, P lies on the opposite from terminal side. opposite from terminal side. Graph the following points on a polar grid. Graph the following points on a polar grid. Complex and Polars.notebook November 18, 2015

Graphing Polar Graphs of polar equations Coordinates • Find four different pairs of polar A polar graph is the set of all points with (r, ) that satisfy a given polar coordinates that name the points equation. 1) Graph r = 2 2) Graph = Check both of these • a) T with your graphing calculator.

• b) S

Keep the three Coordinates Systems Straight!!! Polar Form of a Complex number Rectangular Complex Polar Recall how you graph a + bi. a and b represent the horizontal and vertical components of the point (a, b).

Going to (r, ) | z | "modulus" r = | z| = ______"argument" = ______

Express the following complex numbers in polar form: Rectangular to Polar a) 6 + 8i b) 4 + 3i c) If a point P has rectangular coordinates (x, y) then the polar coordinates ( r, ) of P are given by:

r = and =

Recall, polar coordinates are not unique because we can write one point in multiple ways; however, all rectangular points are unique. Complex and Polars.notebook November 18, 2015

Convert Polar to Rectangular Coordinates Find two polar coordinates for each point with the given rectangular If a point P has polar coordinates (r, ), then the rectangular coordinates. coordinates (x, y) of P are given by 1) (1, 3) 2) ( 3, 6)

x = r cos y = r sin

So, ( x, y) = ( r cos , r sin )

For example, convert (4, pi/6 ) to rectangular form.

Convert the following Polar coordinates to their rectangular form Trig Form of a Complex Number... Since a) b) c) d) x = r cos a y = r sin b for real numbers, the z = a + b i same trig ratios can be = r cos + rsin i applied to find values for = r(cos + sin i ) a and b.... This is the trig form of a complex number!

a) Graph z = 3(cos +i sin ) on a polar grid. Then express it in rectangular form.

c) Complex and Polars.notebook November 18, 2015

Operations with Complex Numbers

Let z1 =

z2 =

Find z1z2 =

Given z1 and z2, Product Formula: Quotient Formula

Example: Find z1z2 and z1/z2 if: Express both answers in polar and rectangular form.

DeMoivre's Theorem If a polar form of a complex number is r(cos + sin i ), then for all positive numbers n:

Example: Find (4 + 4 3 i)6 and express it in rectangular form.

Steps: Find r Find Plug into trig form Apply DM Theorem Complex and Polars.notebook November 18, 2015

Polar Distance Formula

If and are two points in the polar plane, then

the distance P1P2 is given by

Air Traffic: Locations of planes in the sky: Find the distance between (8, 150o) and (3, 65o).