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.