# Thread: Relating to Polar Form (Complex #s, Roots of Unity, DeMoivre's Theorem)

1. ## 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.

2. 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 $\displaystyle l=\sqrt{x^2+y^2}$. x-coordinate of your point X is $\displaystyle l\cos\varphi$ and y-coordinate is $\displaystyle y=l\sin\varphi$. Angle $\displaystyle \varphi$

4. Geometric sequence. Its sum is:
$\displaystyle s_n=a_1\cdot\frac{q^n-1}{q-1}\,;\;q\ne1,n \in\mathbb{N}$

3. 5. That's not hard.
$\displaystyle \begin{array}{rcl}x^5&=&1\\|h^5|(\cos5\varphi+\tex t{i}\sin 5\varphi)&=&1\cdot(\cos0+\text{i}\sin0)\\\end{arra y}$
So $\displaystyle h=1$ and $\displaystyle \cos5\varphi=\cos0\Rightarrow\varphi=\frac{2k\pi}{ 5}\,;\;k\in\mathbb{Z}$
Now, do you know how to carry on?

4. Originally Posted by C^2
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. $\displaystyle x^5 = 1 \Rightarrow x^5 - 1 = 0 \Rightarrow (x - 1)(1 + x + x^2 + x^3 + x^4) = 0 \, ....$

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

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

(1) - (2): $\displaystyle S - r S = a - a r^n \Rightarrow S = \, ....$

5. Originally Posted by mr fantastic
4. $\displaystyle S = a + ar + ar^2 + ... + ar^{n-1}$ .... (1)

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

(1) - (2): $\displaystyle S - r S = a - a r^n \Rightarrow S = \, ....$
what you dont like about formula $\displaystyle s_n=a_n\cdot\frac{q^n-1}{q-1}$ ???

6. Originally Posted by lukaszh
what you dont like about formula $\displaystyle s_n=a_n\cdot\frac{q^n-1}{q-1}$ ???
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.