## kalli one year ago Use W and Z to solve Z= (-5sqrt3)/2+(5/2)i W= 1+(sqrt3) i A. Convert Z and W to polar form

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1. freckles

$r^2=x^2 +y^2 \implies r=\sqrt{x^2+y^2} \\ \tan(\theta)=\frac{y}{x} \\ \text{ do } \theta=\arctan(\frac{y}{x}) \text{ in first and forth quadrants } \\ \text{ do } \theta=\arctan(\frac{y}{x}) +\pi\text{ in second and third quadrants }$

2. freckles

though you can add 2pi *n to those results depending on what range you want theta in

3. kalli

@freckles I think i figured it out by myself .... for Z at least Converted to polar form it would be z=5(cos(5pi/6)+isin(5pi/6) right??

4. freckles

$Z=\frac{-5 \sqrt{3}}{2}+\frac{5}{2}i \\ r=\sqrt{(\frac{-5 \sqrt{3}}{2})^2+(\frac{5}{2})^2}=\sqrt{\frac{25(3)}{4}+\frac{25}{4}}=\sqrt{\frac{25}{4}}\sqrt{3+1}=\frac{5}{2}\sqrt{4}=5$ your r looks amazing

5. freckles

6. kalli

ok :)

7. freckles

$\theta=\arctan(\frac{\frac{5}{2}}{\frac{-5 \sqrt{3}}{2}}) +\pi \\ \theta=\arctan(\frac{5}{2} \cdot \frac{2 }{-5 \sqrt{3}})+ \pi \\ \theta=\arctan(\frac{-1}{\sqrt{3}})+\pi \\ \theta=\frac{-\pi}{6}+\pi$ and your theta looks good as well

8. kalli

Ok, can you help me on W? I got pi/3 but i feel like that may not be right

9. freckles

so since W is in the first quadrant all that you need to do to calculate theta is do arctan(y/x) y is sqrt(3) and x=1

10. freckles

so your theta appears to be right

11. freckles

can you find r

12. kalli

Yay!! Can you help with a second question using these equations? It asks me to convert ZW using De Moivre Theorem but I dont know what de moivre theorem is

13. freckles

$\text{ Say} A=r_1(\cos(\theta_1)+i \sin(\theta_1)) \\ \text{ and } B=r_2(\cos(\theta_2)+i \sin(\theta_2)) \\ AB=r_1 r_2 (\cos(\theta_1+\theta_2)+i \sin(\theta_1+\theta_2))$

14. freckles

15. freckles

when multiplying

16. freckles

if you wanted to do $\frac{A}{B} \\ \text{ well that is } \frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i \sin(\theta_1-\theta_2))$

17. kalli

is that the form im supposed to use?

18. freckles

Well you asked to find AB not A/B I was just sharing that last bit just in case you might find it useful later in the class

19. freckles

And I say you want to use the AB one because you said you want to find ZW

20. freckles

do you understand this: $\text{ Say} A=r_1(\cos(\theta_1)+i \sin(\theta_1)) \\ \text{ and } B=r_2(\cos(\theta_2)+i \sin(\theta_2)) \\ AB=r_1 r_2 (\cos(\theta_1+\theta_2)+i \sin(\theta_1+\theta_2))$ ?

21. kalli

Yeah kind of, I just don't know where to plug in what

22. freckles

your r's for Z and W were 5 and 2 respectively your theta's for Z and W were 5pi/6 and pi/3 respectively your r_1=5 and r_2=2 and your theta_1=5pi/6 and your therta_2=pi/3

23. freckles

$\text{ Say} A=r_1(\cos(\theta_1)+i \sin(\theta_1)) \\ \text{ and } B=r_2(\cos(\theta_2)+i \sin(\theta_2)) \\ AB=r_1 r_2 (\cos(\theta_1+\theta_2)+i \sin(\theta_1+\theta_2))$ $ZW=5\cdot2(\cos(\frac{5\pi}{6}+\frac{\pi}{3})+i \sin(\frac{5\pi}{6}+\frac{\pi}{3}))$

24. freckles

multiply the 5 and 2 there and add the 5pi/6 and the pi/3 there

25. kalli

25 and 3.14??

26. freckles

5(2) is the same as saying 5+5 or 2+2+2+2+2 both of these should add up to be 10

27. freckles

ok hmm...if you aren't sure how to add 5pi/6 and pi/3 maybe just look at 5/6 and 1/3 and that then attach a pi

28. kalli

oooooh lol oops

29. freckles

$\frac{5\pi}{6}+\frac{\pi}{3}=\pi(\frac{5}{6}+\frac{1}{3})=\pi(\frac{5}{6}+\frac{1}{3} \cdot \frac{2}{2}) \\ =\pi(\frac{5}{6}+\frac{2}{6})=?$

30. kalli

pi(5/6+2/6) = 7pi/6??

31. freckles

yeah

32. freckles

$ZW=2 \cdot 5 (\cos(\frac{5\pi}{6}+\frac{\pi}{3})+i \sin(\frac{5\pi}{6}+\frac{\pi}{3})) \\ ZW=10(\cos(\frac{7 \pi}{6})+i \sin(\frac{ 7\pi}{6}))$

33. kalli

Thank you so much! Do you think you could help with one last part to this?? Its similar to what we just did so mybe ill have a better understanding

34. freckles

ok

35. kalli

C. Calculate z/w using de moivre theorem

36. freckles

ok well actually I mentioned that quotient above with A and B

37. freckles

let me copy and paste one sec

38. freckles

$\text{ Say} A=r_1(\cos(\theta_1)+i \sin(\theta_1)) \\ \text{ and } B=r_2(\cos(\theta_2)+i \sin(\theta_2))$ $\frac{A}{B} \\ \text{ well that is } \frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i \sin(\theta_1-\theta_2))$

39. freckles

Identify your r1 and r2 and theta1 and theta2 and plug in the formula for A/B

40. freckles

where A=Z and B=W in this case

41. kalli

are the r's still 5 and 2?

42. freckles

yep because we are still using Z And W

43. kalli

and my thetas are 5pi/6 and pi/3 yes?

44. freckles

yep

45. kalli

(cos(5pi/6 -5pi/6) + isin(pi/3-pi/3) ??

46. freckles

for the inside you are suppose to do Z's theta - W's theta and you still need Z's r / W's r on the outside of the ( )

47. freckles

$\text{ Say} A=r_1(\cos(\theta_1)+i \sin(\theta_1)) \\ \text{ and } B=r_2(\cos(\theta_2)+i \sin(\theta_2))$ $\frac{A}{B} \\ \text{ well that is } \frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i \sin(\theta_1-\theta_2))$ do you how for A/B on the outside you have A's r divided by B's r and the angles inside the cos and sin thing is A's theta- B's theta?

48. kalli

no im so confused right now lol

49. freckles

do you understand in that formula r1 is A's radius?

50. freckles

and that r2 is B's?

51. freckles

and that theta1 is A's angle thing and theta2 is B's angle thing?

52. freckles

you are doing Z/W so your r1 and theta1 will come from the information you got from Z and r2 and theta2 will come from the information you got from W

53. freckles

earlier you told me r1=5 and r2=2 and theta1=5pi/6 and theta2=pi/3 just plug in the formula

54. freckles

$\text{ Say} A=r_1(\cos(\theta_1)+i \sin(\theta_1)) \\ \text{ and } B=r_2(\cos(\theta_2)+i \sin(\theta_2))$ $\frac{A}{B} \\ \text{ well that is } \frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i \sin(\theta_1-\theta_2))$ $Z=5(\cos(\frac{5 \pi}{6})+ i \sin(\frac{5 \pi}{6})) \\ W=2 (\cos(\frac{\pi}{3})+i \sin(\frac{\pi}{3})) \\ r_1=5,r_2=2 ,\theta_1=\frac{5\pi}{6}, \theta_2=\frac{\pi}{3} \\ \text{ plug into the formula } \\ \frac{Z}{W}=\frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i \sin(\theta_1-\theta_2))$

55. freckles

replace the r1 with 5 replace the r2 with 2 theta1 with 5pi/6 theta 2 with pi/3

56. kalli

5(1)/2(2) = (cos(5pi/6-pi/3) +isin(5pi/6-pi/3) ??

57. freckles

ok but there shouldn't be an equal sign in between r1/r2 and (cos(thet...blah blah) also where does the extra 1 and 2 come from?

58. kalli

oh wait i misunderstood its suppose to be 5/2

59. freckles

yep

60. freckles

and also remember to omit the equal sign

61. freckles

between the r1/r2 and the (cos...blah blah)

62. kalli

5/2(cos(5pi/6-pi/3) +isin(5pi/6-pi/3) ??

63. freckles

yep

64. kalli

65. freckles

well just like we did 5pi/6+pi/3 earlier you could do 5pi/6-pi/3

66. kalli

pi/2

67. freckles

$\frac{5\pi}{6}-\frac{\pi}{3}=\pi(\frac{5}{6}-\frac{1}{3})=\pi(\frac{5}{6}-\frac{2}{6})=\pi(\frac{3}{6})=\pi \frac{1}{2}= \frac{\pi}{2}$

68. freckles

oh yes

69. freckles

5/2 can be written as 2.5 but I think it looks just as pretty as 5/2