## tdabboud Group Title If z= (-2+7i) and w = z^3000 whats the angle theta of w in the region [0,2pi) I want to know how to compute this. I can easily plug it into wolfram but it doesnt explain the steps 2 years ago 2 years ago

1. tdabboud Group Title

would it simply be 3000 * (angle of (-2+7i)) ?

2. EarthCitizen Group Title

nope

3. EarthCitizen Group Title

$w=(-2+j7)^{3000}$

4. EarthCitizen Group Title

there's two way's to solve this either directly or use de Moivre's theorem, the choice is yours

5. tdabboud Group Title

i havent learned that theorem yet so most likely the direct way

6. EarthCitizen Group Title

the direct way is log as you can see the power is 3000

7. tdabboud Group Title

yea but we're dealing with complex numbers.. do the same rules apply then

8. EarthCitizen Group Title

first thing you need to do here is change z into polar form

9. EarthCitizen Group Title

that is in the form of $r ^{n}<n \theta$

10. tdabboud Group Title

ok then...(7.28,1.85)

11. JamesJ Group Title

@EC, that's not polar form for complex numbers. You want to write $z = Ae^{i\theta}$ for some positive number $$A$$ which is the magnitude of $$z$$, and some angle $$\theta \in [0,2\pi)$$.

12. JamesJ Group Title

Then $z^{3000} = A^{3000} e^{3000\theta i}$

13. EarthCitizen Group Title

yh, in polar form $r<\theta$

14. JamesJ Group Title

(@EC, I see. Not standard notation, but I get it.)

15. EarthCitizen Group Title

@ JamesJ is Euler's theorem the only way to solve ?

16. tdabboud Group Title

i didnt think it was the only way, there has to be another

17. JamesJ Group Title

It's far away the fastest. Unless you use some version of it, you would need to multiply that out explicitly. The whole point of the exercise with such a ridiculous exponent is to make you use some polar form.

18. JamesJ Group Title

Do you not have deMoivre's theorem?

19. EarthCitizen Group Title

yh couldn't we just say$(r ^{n}<n \theta)$

20. tdabboud Group Title

yea i do understand that, the coordinates i gave were correct for eulers and polar: w= 7.28 e^j(1.85 rad)

21. tdabboud Group Title

using eulers you would then take the log?

22. EarthCitizen Group Title

$e ^{j \theta}= \cos(\theta)+jsin (\theta)$

23. tdabboud Group Title

I am still lost... one of my problems we did they simply took the exponent to the front and multiplied it by the angle the problem was (-3+j)^4 so they did: 4(2.8198 rad) = 11.27 rad... you can then subtract 2pi to get to the correct domain. b but i tried this and its not working correctly

24. EarthCitizen Group Title

$(re ^{j \theta})=r ^{n}e ^{jn \theta}=(7.28)^{3000}(\cos(3000\theta)+jsin(3000\theta)$

25. EarthCitizen Group Title

to convert to radians$(\theta \times \pi)/180$

26. EarthCitizen Group Title

that's de Moivre's, $(r<\theta)^{n}=(r ^{n}<n \theta)$

27. EarthCitizen Group Title

$\theta=105.95$

28. EarthCitizen Group Title

29. tdabboud Group Title

the angle you gave 1.85 rad is only for (-2+7j)^1

30. tdabboud Group Title

but when raised to the 3000 the angle becomes -.765

31. EarthCitizen Group Title

i know there're 2999 more

32. tdabboud Group Title

yeah thats the problem I am having i found the angle you gave me easily, but how can you find that angle when raised to the 3000. would it be 3000 times that angle like we have done previously? then just reduce it to 0,2pi

33. EarthCitizen Group Title

are you looking for a specific angle ?

34. tdabboud Group Title

yeah an angle reduced to the domain between 0 and 2pi. or between 0 and 360 degrees.

35. EarthCitizen Group Title

which angle there're 3000 of them ?

36. tdabboud Group Title

haha the one between 0 and 360 degrees, there will be only one. the other angles have a greater total degree

37. EarthCitizen Group Title

no the angles can't exceed 360 degrees

38. EarthCitizen Group Title

all 3000 of them are within the range of 0 to 360

39. EarthCitizen Group Title

you can use this formula to find one angle between the range of 0 to 360 $r ^{1/n}<(\theta+360(k))/n$

40. EarthCitizen Group Title

n= power, k=position of the angle

41. EarthCitizen Group Title

http://www.wolframalpha.com/input/?i=%28-2%2Bi7%29^3000 these are all the angles according to wolfy