Question 302205: Find the cube roots of 2 + i 2 . Locate them graphically
Found 2 solutions by Alan3354, stanbon:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! Find the cube roots of 2 + i 2 . Locate them graphically
z = sqrt(8)cis(45)
z^(1/3) = (8^(1/6))cis(15)
= sqrt(2)cis(15)
= sqrt(2)[cos(15,135,255) + isin(15,135,255)]
---------------------
You can use the half-angle formulas to eliminate the sine and cosine.
Answer by stanbon(75887)  (Show Source):
You can put this solution on YOUR website! Find the cube roots of 2 + i 2 . Locate them graphically
-----------------------
Plot the point (2,2) so you can see what is happening.
-----
Convert to trip form:
r = sqrt(2^2 + 2^2)= 2sqrt(2) = 8^(1/2)
theta = tan^-1(2/2) = 45 degrees + 360n
-----------
(2+2i)^(1/3) = (8)^(1/6)cis[(45+360n)/3] for n = 0,1,2
---
If n = 0 ; 8^(1/6)cis(15)
If n = 1 ; 8^(1/6)cis(135)
If n = 2 ; 8^(1/6)cis(255)
----
Graphically these are three points on a circle centered
at the origin with radius = 8^(1/6).
The points are located at 15 degrees, 135 degrees, and 255 degrees
===================================================================
Cheers,
Stan H.
================
|
|
|
