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    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
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    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
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    (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?
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    (Original post by Manifest)
    Please solve any example or do you want me to give one?
    example?
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    how do I find nth roots of unity?
    Check page 10
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    (Original post by Dekota)
    Check page 10
    Thanks..
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    (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]
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    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.
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    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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