I need some help with the following questions:
1) Let z and w be the complex numbers in the picture below:
If |z|=4 and |w|=2 then what is the imaginary part of z/w?
2) Consider the complex numbers
and
in the following picture, with the various angles between the line segments marked:
Here, we have that
and
. Then calculate the ordered list
![\[ z_1^6, z_2^6, z_3^6, z_4^6.\]](https://latex.artofproblemsolving.com/1/9/e/19e8fa9f6394e190838f58982902477f9a8dc55b.png "\[ z_1^6, z_2^6, z_3^6, z_4^6.\]")
Enter in your answers in order, in rectangular form.
3) The equation
![\[z^3 = -2 - 2i\]](https://latex.artofproblemsolving.com/9/7/8/978566a4c03da64e9abfc506b8df9b63f0f9f8e6.png "\[z^3 = -2 - 2i\]")
has
solutions. What is the unique solution in the fourth quadrant? Enter in your answer in rectangular form.
4) There exist two complex numbers
, say
and
, so that
,
, and
form the vertices of an equilateral triangle. Find the product
in rectangular form.
5) Consider the curve with equation
![\[\operatorname{Re}\left( \dfrac{1}{z} \right) = \dfrac{1}{6}.\]](https://latex.artofproblemsolving.com/7/a/0/7a00868de42f518de56453793ab109c6adddd8cc.png "\[\operatorname{Re}\left( \dfrac{1}{z} \right) = \dfrac{1}{6}.\]")
For each complex number in the following list,
![\[1, 4i, 3+3i, 3-3i, 1 - 2i, 2+ 3i, 6,\]](https://latex.artofproblemsolving.com/b/f/3/bf32f8aeaf760aadd1b0c6a2b90f91392be5eebe.png "\[1, 4i, 3+3i, 3-3i, 1 - 2i, 2+ 3i, 6,\]")
figure out whether each one is on the curve, then enter "yes" or "no" in the blank corresponding to each option below.
1) Let z and w be the complex numbers in the picture below:
If |z|=4 and |w|=2 then what is the imaginary part of z/w?
2) Consider the complex numbers
Here, we have that
3) The equation
4) There exist two complex numbers
5) Consider the curve with equation
For each complex number in the following list,
