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# De Moivre's Theorem? watch

1. Could anyone please tell me how to express tan5θ in term of powers of tanθ.
I can actually express sinθ and cosθ and I know sinθ/cosθ is tanθ but I can finish up. I hope you guys get what I mean?

Secondly, how do I find nth roots of unity? It is confusing, my text book isn't explaining clearly and I have exams tommorrow... I have used all my time revising for C3 and C4...Urgent help needed. Might REP ...thanks in advance.

Manifest
2. tan5θ into powers of tanθ

I don't know if you can do it easily with De Moivre, which relates cosθ and sinθ.... I could be wrong though, but since tanθ is a quotient, I can't immediately see an obvious way of doing it.

Anyway, nth roots of unity:

zn = 1

set 1 = e2πi*k (k is any integer)

zn = e2πi*k

implies

z = e2πi*(k/n)

implies kth root is e2πi*(k/n)....
of course, for k>n, you've come full circle, so the only roots you write down are

e2πi(1/n), e2πi(2/n) .... e2πi(k/n) .... e2πi((n-1)/n), e2πi = 1

Hope this helps
3. (Original post by Olek)
tan5θ into powers of tanθ

I don't know if you can do it easily with De Moivre, which relates cosθ and sinθ.... I could be wrong though, but since tanθ is a quotient, I can't immediately see an obvious way of doing it.

Anyway, nth roots of unity:

zn = 1

set 1 = e2πi*k (k is any integer)

zn = e2πi*k

implies

z = e2πi*(k/n)

implies kth root is e2πi*(k/n)....
of course, for k>n, you've come full circle, so the only roots you write down are

e2πi(1/n), e2πi(2/n) .... e2πi(k/n) .... e2πi((n-1)/n), e2πi = 1

Hope this helps
Maybe this is where I got confused...is it k/n or everything over n?
Please solve any example or do you want me to give one?
4. (Original post by Manifest)
Please solve any example or do you want me to give one?
example?
5. how do I find nth roots of unity?
Check page 10
6. (Original post by Dekota)
Check page 10
Thanks..
7. (Original post by Manifest)
Could anyone please tell me how to express tan5θ in term of powers of tanθ.
I can actually express sinθ and cosθ and I know sinθ/cosθ is tanθ but I can finish up. I hope you guys get what I mean?

Secondly, how do I find nth roots of unity? It is confusing, my text book isn't explaining clearly and I have exams tommorrow... I have used all my time revising for C3 and C4...Urgent help needed. Might REP ...thanks in advance.

Manifest
this is probably what you have but just in case:
cos 5θ+isin5θ=(cos θ+isinθ )^5
=c^5+5c^4is+10c^3i^2s^2+10c^2i^3 s^3+5ci^4s^4+i^5s^5
=c^5-10c^3s^2+5cs^4+i[5c^4s-10c^2s^3+s^5]
comparing real and imaginary parts leads to
cos5θ=cos^5 θ-10cos³θsin²θ+5cosθsin^4 θ
sin 5θ=5cos^4 θsinθ-10cos²θsin³θ+sin^5 θ

hence
cos5θ=cos^5 θ[1-tan²θ+5tan^4 θ]
sin5θ=cos^5 θ[5tanθ-10tan³θ+tan^5 θ]

tan5θ=[5tanθ-10tan³θ+tan^5 θ]/[1-tan²θ+5tan^4 θ]
8. Thanks....

as for the unity...How do you find other angles? is it in all cases you increase the value of k by 1? I saw an example that says w³ = 1 and k was btw -1, 0 and 1.
9. basically, if you take the nth root, there are n roots....

follow my working, you'll see that including 1, there are n roots to the equation.

I don't understand where you get the k = -1, 0 , 1 from

literally, apply my solution directly and draw the argand diagram and you will see....

a couple of obvious solutions

2nd roots of unity:

e2iπ(1/2) = e = -1
e2iπ = 1

roots are -1 and 1

4th roots of unity:

e2iπ(1/4) = eiπ/2 = i
e2iπ(2/4) = e = -1
e2iπ(3/4) = eiπ(3/2) = -i
e2iπ(4/2) = e2iπ = 1

you get the idea.... 3rd roots of unity are just 1/3 of the way round the argand diagram...

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Updated: January 27, 2006
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