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anonymous
 one year ago
*WILL FAN AND MEDAL* Can someone please take a look at my work and help me work out the rest of this problem?
Find the cube roots of 27(cos 330° + i sin 330°).
anonymous
 one year ago
*WILL FAN AND MEDAL* Can someone please take a look at my work and help me work out the rest of this problem? Find the cube roots of 27(cos 330° + i sin 330°).

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[z = r(\cos \theta + i \sin \theta)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\sqrt[n]{r}(\cos \frac{ \theta+2\pi k }{ n }+i \sin \frac{ \theta+2\pi k }{ n })\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I substituted the values into the formula above. K is any real number so I chose to use k= 0, 1, 2. For my first root working with k=0, I got \[3(\cos \frac{ \theta }{ 3 }+i \sin \frac{ \theta }{ 3 })\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Im stuck on finding the second and third roots..

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{330}{3}=111\] right?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2since you seem to be working in degrees, which is weird, but whatever

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2then to get the next one, go around the circle again

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2\[330+360=690\] and \[\frac{690}{3}=230\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2so next angle is \(230\) then go around yet again

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2i can see why it is confusing, because you are using that \(\frac{\theta+2k\pi}{n}\) thing, but you using degrees

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2if you were using radians, then that is what you would use but since you are using degrees, keep adding 360 then divide not \(2\pi\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Im using DeMoivre's Theorem, but I was just confused because I'm not sure if I should substitute the angle 330 degrees in place of theta in the expression I have so far. In finding my second root, I used my second value for k, which is 1. Im just stuck on whether or not im supposed to substitute in 330 degrees in place of theta. This is the work I have built around my second root: \[\sqrt[3]{27}(\cos \frac{ \theta+2 \pi(1)}{ 3 }+ i \sin \frac{ \theta+2 \pi(1) }{ 3})\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I cant give my answers in degrees on this problem though.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Because the standard form of the expression i have is \[z=r(\cos \theta + i \sin \theta)\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2if you are working in degrees, then you have to add 360 not \(2\pi\)

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2and the problem was given in degrees, so probably you are supposed to answer in degree otherwise it is just too weird

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2if you want to work in radians, then you would need to convert \(330\) in to radians as step one

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0For my first root I got \[3(\cos \frac{ \theta }{ 3 } +i \sin \frac{ \theta }{ 3 })\] So if i am to give my answer in degree form, how do I get that from this answer?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2\(\theta\) is the angle in this case \(\theta=330\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Alright, so i substitute 330 in place of theta and simplify?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2looks like you may be confusing yourself with the formula in simple english to take the cubed root, divide the angle by 3

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2so first answer is \[3\left(\cos(110^\circ)+i\sin(110^\circ)\right)\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2to get the second answer, as i wrote above, add \(360\) to \(330\) then divide by 3 IF you were working in radians (which you are not) then you would add \(2\pi\) but since you are working in degrees, add \(360\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I see what you mean here by how confusing it is. So I could solve this just by using a unit circle and finding coterminal angles and dividing them by 3 to get cube roots?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2yeah just keep going around

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2don't forget for the trig form of a complex number the angle is not unique, since sine and cosine are periodic

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Okay, so my first root would be 330/3= 110?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2no, dear, that is your first ANGLE the root is \[3\left(\cos(110^\circ)+i\sin(110^\circ)\right)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0OH! Okay, so I do that process but just substitute in my answer for theta?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2you have no idea what that number is, since you know nether \(\cos(110^\circ)\) or \(\sin(110^\circ)\) just leave it in that form

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2ready to find the next root? there are three total

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Right, thats what I mean. So my second root would look like this: \[3(\cos(230)+i \sin (230))\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Oh, and then my last root would look like this: 3(cos(196.6)+isin(196.6))

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2lets back up a second first angle was \(330\div 3=110\)

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2second one was \(\frac{330+360}{3}=230\)

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2third one add 360 again \[\frac{330+360+360}{3}\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2or more to your formula \[\frac{330+2\times 360}{3}\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2you keep going around the circle

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2not sure where the 196.6 came from

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Oh my gosh, thank you so much for explaining that. I really needed it. You are by far the most persistent person yet. Thank you so much!

misty1212
 one year ago
Best ResponseYou've already chosen the best response.2\[\huge \color\magenta\heartsuit\] btw i hope it is now clear who easy this is

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Crystal clear, thank you!
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