A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

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°).

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    My work so far:

  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[z = r(\cos \theta + i \sin \theta)\]

  3. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\sqrt[n]{r}(\cos \frac{ \theta+2\pi k }{ n }+i \sin \frac{ \theta+2\pi k }{ n })\]

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I 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 \[3(\cos \frac{ \theta }{ 3 }+i \sin \frac{ \theta }{ 3 })\]

  5. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Im stuck on finding the second and third roots..

  6. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    HI!!

  7. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[\frac{330}{3}=111\] right?

  8. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Correct @misty1212

  9. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    since you seem to be working in degrees, which is weird, but whatever

  10. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    then to get the next one, go around the circle again

  11. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[330+360=690\] and \[\frac{690}{3}=230\]

  12. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    so next angle is \(230\) then go around yet again

  13. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    i can see why it is confusing, because you are using that \(\frac{\theta+2k\pi}{n}\) thing, but you using degrees

  14. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    if you were using radians, then that is what you would use but since you are using degrees, keep adding 360 then divide not \(2\pi\)

  15. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Im 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: \[\sqrt[3]{27}(\cos \frac{ \theta+2 \pi(1)}{ 3 }+ i \sin \frac{ \theta+2 \pi(1) }{ 3})\]

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I cant give my answers in degrees on this problem though.

  17. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Because the standard form of the expression i have is \[z=r(\cos \theta + i \sin \theta)\]

  18. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    if you are working in degrees, then you have to add 360 not \(2\pi\)

  19. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    and the problem was given in degrees, so probably you are supposed to answer in degree otherwise it is just too weird

  20. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    if you want to work in radians, then you would need to convert \(330\) in to radians as step one

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    For 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?

  22. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh ok

  23. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \(\theta\) is the angle in this case \(\theta=330\)

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Alright, so i substitute 330 in place of theta and simplify?

  25. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    330/3 = 110

  26. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    looks like you may be confusing yourself with the formula in simple english to take the cubed root, divide the angle by 3

  27. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    so first answer is \[3\left(\cos(110^\circ)+i\sin(110^\circ)\right)\]

  28. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    to 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 \(360\)

  29. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I 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?

  30. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    yeah just keep going around

  31. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    don't forget for the trig form of a complex number the angle is not unique, since sine and cosine are periodic

  32. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Okay, so my first root would be 330/3= 110?

  33. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    no, dear, that is your first ANGLE the root is \[3\left(\cos(110^\circ)+i\sin(110^\circ)\right)\]

  34. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    OH! Okay, so I do that process but just substitute in my answer for theta?

  35. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    you have no idea what that number is, since you know nether \(\cos(110^\circ)\) or \(\sin(110^\circ)\) just leave it in that form

  36. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    ready to find the next root? there are three total

  37. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Right, thats what I mean. So my second root would look like this: \[3(\cos(230)+i \sin (230))\]

  38. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    exactly!

  39. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh, and then my last root would look like this: 3(cos(196.6)+isin(196.6))

  40. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    hmmm no

  41. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    lets back up a second first angle was \(330\div 3=110\)

  42. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes

  43. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    second one was \(\frac{330+360}{3}=230\)

  44. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes

  45. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    third one add 360 again \[\frac{330+360+360}{3}\]

  46. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    or more to your formula \[\frac{330+2\times 360}{3}\]

  47. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    350

  48. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    you keep going around the circle

  49. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    right

  50. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    not sure where the 196.6 came from

  51. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh 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!

  52. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[\huge \color\magenta\heartsuit\] btw i hope it is now clear who easy this is

  53. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Crystal clear, thank you!

  54. misty1212
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    you're welcome dear

  55. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.