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Math Help - Relating to Polar Form (Complex #s, Roots of Unity, DeMoivre's Theorem)

  1. #1
    C^2
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    Relating to Polar Form (Complex #s, Roots of Unity, DeMoivre's Theorem)

    I have some questions on a worksheet dealing with polar form, complex #s, and roots of unity. Any help on any of these problems will be much appreciated.

    1. Choose a point in the complex plane, and find four different ways of using polar coordinates to describe it, using angles between -2pie and 2 pie. Use your work to describe a fundamental difference between rectangular and polar coordinates.

    2. Use the angle addition identity cisAcisB = cis(A+B) to prove DeMoivre's theorem: (rcisX)^n = (r^n)(cisnX)

    3. Multiply a +bi by c +di for the special case when a=cosX, b=sinX, c=cosY, and d=sinY. Use your result to prove the angle-addition property of complex multiplication. Explain how this idea could help you reconstruct certain formulas.

    4. Simplify a + ar + ar^2 + ... + ar^(n-1). (Write it without dots)

    5. A solution to the equation x^n=1 is called an nth root of unity. If x is an nth root of unity and x^k dies not equal 1 for all integers k, 0<k<n, then x is called a primitive nth root of unity. Use DeMoivre's Theorem to find a primitive 5th root of unity. Call it x, and compute 1+x+x^2+x^3+x^4. Find a geometric explanation for your answer.
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    You should know what is polar coordinate system. If you have a point in plane, and you connect [0,0] with your point [x,y] its a vector. Its length is l=\sqrt{x^2+y^2}. x-coordinate of your point X is l\cos\varphi and y-coordinate is y=l\sin\varphi. Angle \varphi

    4. Geometric sequence. Its sum is:
    s_n=a_1\cdot\frac{q^n-1}{q-1}\,;\;q\ne1,n \in\mathbb{N}
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    5. That's not hard.
    \begin{array}{rcl}x^5&=&1\\|h^5|(\cos5\varphi+\tex  t{i}\sin 5\varphi)&=&1\cdot(\cos0+\text{i}\sin0)\\\end{arra  y}
    So h=1 and \cos5\varphi=\cos0\Rightarrow\varphi=\frac{2k\pi}{  5}\,;\;k\in\mathbb{Z}
    Now, do you know how to carry on?
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  4. #4
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    Quote Originally Posted by C^2 View Post
    I have some questions on a worksheet dealing with polar form, complex #s, and roots of unity. Any help on any of these problems will be much appreciated.

    1. Choose a point in the complex plane, and find four different ways of using polar coordinates to describe it, using angles between -2pie and 2 pie. Use your work to describe a fundamental difference between rectangular and polar coordinates.

    2. Use the angle addition identity cisAcisB = cis(A+B) to prove DeMoivre's theorem: (rcisX)^n = (r^n)(cisnX)

    3. Multiply a +bi by c +di for the special case when a=cosX, b=sinX, c=cosY, and d=sinY. Use your result to prove the angle-addition property of complex multiplication. Explain how this idea could help you reconstruct certain formulas.

    4. Simplify a + ar + ar^2 + ... + ar^(n-1). (Write it without dots)

    5. A solution to the equation x^n=1 is called an nth root of unity. If x is an nth root of unity and x^k dies not equal 1 for all integers k, 0<k<n, then x is called a primitive nth root of unity. Use DeMoivre's Theorem to find a primitive 5th root of unity. Call it x, and compute 1+x+x^2+x^3+x^4. Find a geometric explanation for your answer.
    5. x^5 = 1 \Rightarrow x^5 - 1 = 0 \Rightarrow (x - 1)(1 + x + x^2 + x^3 + x^4) = 0 \, ....


    4. S = a + ar + ar^2 + ... + ar^{n-1} .... (1)

    r S = ar + ar^2 + ar^3 + ... + ar^n .... (2)

    (1) - (2): S - r S = a - a r^n \Rightarrow S = \, ....
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    Quote Originally Posted by mr fantastic View Post
    4. S = a + ar + ar^2 + ... + ar^{n-1} .... (1)

    r S = ar + ar^2 + ar^3 + ... + ar^n .... (2)

    (1) - (2): S - r S = a - a r^n \Rightarrow S = \, ....
    what you dont like about formula s_n=a_n\cdot\frac{q^n-1}{q-1} ???
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    Quote Originally Posted by lukaszh View Post
    what you dont like about formula s_n=a_n\cdot\frac{q^n-1}{q-1} ???
    1. I prefer to give answers that use the same pronumerals as those used by the OP.

    2. I suspect a derivation of the sum was required, rather than simply quoting the formula.
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