Question

What is −√3−i in polar form, i.e. write as: r * cis(θ)
for some real r and θ... help please!

Answer 1

x+yi is in rectangular form

r*cis(theta) is in polar form

To convert, we use the equations

x^2+y^2 = r^2 ----> r = sqrt(x^2+y^2)

and

theta = arctan(y/x)

------------------------------------------------

In our case, x = -sqrt(3) and y = -1

So r = sqrt( (-sqrt(3))^2 + (-1)^2) = sqrt( 3 + 1) = sqrt(4) = 2

Giving us r = 2

and

theta = arctan(-1/(-sqrt(3)) = arctan( sqrt(3)/3 ) = pi/6

So theta = pi/6


This means that -sqrt(3) - i converts to 2*cis(pi/6)

which can be expanded out to get 2*(cos(pi/6) + i*sin(pi/6))

Answer 2

Oh I made a typo...

The point -sqrt(3) - i is in the third quadrant. So arctan(-1/(-sqrt(3))) will be in the third quadrant which means you have to add pi to the angle theta


So theta is really pi/6 + pi = 7pi/6

Answer 3

You are a champion, thank you so much!

Answer 4

you're welcome

Answer 5

Hmm this is for an online maple quiz, and when I put in that answer, I get the following:
"Your answer is partially correct.
Hint: sketch −√3−i on the Argand plane, and use geometry to discover its modulus and Arg."
Could you help me with what this hint means?

Answer 6

I thought thats what you did, ie, using pythagoras to get r, etc...

Answer 7

That's exactly right. You can think of any point in the complex plane as some point that forms a right triangle (where one leg lies on the x axis and the right angle sits on the x axis). The distance that point is from the origin is found by the expression sqrt(x^2+y^2)


But this distance also describes the radius of a circle that is centered at the origin and passes through this point. This is where polar form then comes into play. So that explains why the conversion works the way it does.

Answer 8

Perhaps the angle needs to be in degrees?

Answer 9

OK thanks, I get that now. But what about the error message I got on the question? What makes cis(7pi/6) incorrect?

Answer 10

it should be 2*cis(7pi/6)

Answer 11

I don't think it would be in degrees.... I get 0.9/1 for the questions using that answer.... It's strange :/

Answer 12

assuming that the angle is in radians

Answer 13

Yeah I have added the 2* as well, sorry, typo

Answer 14

so it's saying that the answer is r = 0.9 and theta = 1? I'm not following what you mean on that piece.

Answer 15

Oh sorry - a mark of 90% is what I meant. Implying that there is something rather small which is incorrect, related to the "Hint" which I posted earlier.. Does that make more sense?

Answer 16

Oh ok, let me reread the hint real quick

Answer 17

The only thing I can think of is that the angle is supposed to be -5pi/6

Why -5pi/6? I'm going with this because wolfram alpha is spitting this answer out and because

pi/6 - pi = pi/6 - 6pi/6 = -5pi/6


But that seems trivial since it's coterminal to 7pi/6

Answer 18

Let me know what the system says if you try that answer.

Answer 19

Yeah it liked that answer... How pedantic.. Haha cheers for the help, as always!

Answer 20

Umm ok, so the next part of the question, if its ok to keep bothering you with this.... is "Now rewrite (−√3−i)i using −√3−i in polar form" ... do you just do the same thing but backwards? does it have something to do with 2cis(-5pi/6)=2exp(i*(-5pi/6))?

Answer 21

First convert i to polar form. It might help to realize that i = 0+1i


So x = 0 and y = 1 which means r = sqrt(0^2+1^2) = sqrt(1) = 1, so r = 1

and theta = pi/2 ... Note: you can't use the formula posted above to find theta because you'll be dividing by zero, but the good news is that this angle is easy to see/figure out.


So 0+1i converts to 1*cis(pi/2)


So...


(-sqrt(3) - i)i = [2*cis(-5pi/6)]*[1*cis(pi/2)] = (2*1)*cis(-5pi/6+pi/2) = 2*cis(-2pi/6) = 2*cis(-pi/3)


So (-sqrt(3) - i)i converts to 2*cis(-pi/3)

Answer 22

Hmmm okay, I see what you've done there, but I get a comment saying "remember to use use arg rather than Arg. So this will introduce an integer parameter; please use k." .. which seems unrelated :/ sorry to keep bugging you, let me know if you can't help any more haha

Answer 23

Oh so they want all angles that coincide with -pi/3?


If that's the case, then simply add multiples of 2pi to the angle to get -pi/3 + 2pi*k

Answer 24

where k is any integer

Answer 25

Ohh I see! Thanks!

Answer 26

Sure thing

Answer 27

Hahaha wow how sick of me are you? That answer is still incorrect... Oh well.. I'm just about at giving up point!!

Answer 28

hmm maybe they want 5pi/3 since


-pi/3 + 2pi = -pi/3 + 6pi/3 = 5pi/3

Answer 29

No thats not it either... Oh well... :S I think I might just speak to our tutor about it in tomorrows class! Thanks so much for all you help though - I understand the concepts way more now than I did an hour ago!

Answer 30

That's great. It's a shame that the computer has to be fussy about the final answer though...

Answer 31

Yeah it's the main reason I dislike these online quiz assessments... You also get deducted 0.1 out of 1 for every incorrect answer you input, and it obviously doesn't account for typo's or incorrect forms.. .Oh well! Counts as study, so I shouldn't be minding really :) cheers again!

Answer 32

I see, well at least you're thinking about different possible solutions and how they all connect (ie you're getting a view of the bigger picture)