Question

Please HELP!!!!

Answer 1

Answer:

Step-by-step explanation:

Firstly, note that -2i really is just z = 0 + (-2)i, so we see that Re(z) = 0 and Im(z) = -2.

When we're going from Cartesian to polar coordinates, we need to be aware of a few things! With Cartesian coordinates, we are dealing explicitly with x = blah and y = blah. With polar coordinates, we are looking at the same plane but with angle and magnitude in consideration.

Graphing z = -2i on the Argand diagram will look like a segment of the y axis. So we ask ourselves "What angle does this make with the positive x axis? One answer you could ask yourself is -90°! But at the same time, it's 270°! Why do you think this is the case?

What about the magnitude? How far is "-2i" stretched from the typical "i". And the answer is -2! Well... really it gets stretched by a factor of 2 but in the negative direction!

Putting all of this together gives us:

z = |mag|*(cos(angle) + isin(angle))

= 2*cos(270°) + isin(270°)).

To verify, let's consider what cos(270°) and sin(270°) are.

If you graph cos(x) and look at 270°, you get 0.

If you graph sin(x) and look at 270°, you get -1.

So 2*(cos(270°) + isin(270°)) = 2(0 + -1*i) = -2i as expected.