 8.4.1: Fill in the blanks to correctly complete each problem. If z = 31cos...
 8.4.2: Fill in the blanks to correctly complete each problem. If we are gi...
 8.4.3: Fill in the blanks to correctly complete each problem. 3cos 6 + i s...
 8.4.4: Fill in the blanks to correctly complete each problem. Based on the...
 8.4.5: How many real tenth roots of 1 exist?
 8.4.6: How many nonreal complex tenth roots of 1 exist?
 8.4.7: Find each power . Write answers in rectangular form. See Example 1....
 8.4.8: Find each power . Write answers in rectangular form. See Example 1....
 8.4.9: Find each power . Write answers in rectangular form. See Example 1....
 8.4.10: Find each power . Write answers in rectangular form. See Example 1....
 8.4.11: Find each power . Write answers in rectangular form. See Example 1....
 8.4.12: Find each power . Write answers in rectangular form. See Example 1....
 8.4.13: Find each power . Write answers in rectangular form. See Example 1....
 8.4.14: Find each power . Write answers in rectangular form. See Example 1....
 8.4.15: Find each power . Write answers in rectangular form. See Example 1....
 8.4.16: Find each power . Write answers in rectangular form. See Example 1....
 8.4.17: Find each power . Write answers in rectangular form. See Example 1....
 8.4.18: Find each power . Write answers in rectangular form. See Example 1....
 8.4.19: For each of the following, (a) find all cube roots of each complex ...
 8.4.20: For each of the following, (a) find all cube roots of each complex ...
 8.4.21: For each of the following, (a) find all cube roots of each complex ...
 8.4.22: For each of the following, (a) find all cube roots of each complex ...
 8.4.23: For each of the following, (a) find all cube roots of each complex ...
 8.4.24: For each of the following, (a) find all cube roots of each complex ...
 8.4.25: For each of the following, (a) find all cube roots of each complex ...
 8.4.26: For each of the following, (a) find all cube roots of each complex ...
 8.4.27: For each of the following, (a) find all cube roots of each complex ...
 8.4.28: For each of the following, (a) find all cube roots of each complex ...
 8.4.29: For each of the following, (a) find all cube roots of each complex ...
 8.4.30: For each of the following, (a) find all cube roots of each complex ...
 8.4.31: Find and graph all specified roots of 1.second (square)
 8.4.32: Find and graph all specified roots of 1.fourth
 8.4.33: Find and graph all specified roots of 1. sixth
 8.4.34: Find and graph all specified roots of i.second (square)
 8.4.35: Find and graph all specified roots of i.third (cube)
 8.4.36: Find and graph all specified roots of i.fourth
 8.4.37: Find all complex number solutions of each equation. Write answers i...
 8.4.38: Find all complex number solutions of each equation. Write answers i...
 8.4.39: Find all complex number solutions of each equation. Write answers i...
 8.4.40: Find all complex number solutions of each equation. Write answers i...
 8.4.41: Find all complex number solutions of each equation. Write answers i...
 8.4.42: Find all complex number solutions of each equation. Write answers i...
 8.4.43: Find all complex number solutions of each equation. Write answers i...
 8.4.44: Find all complex number solutions of each equation. Write answers i...
 8.4.45: Find all complex number solutions of each equation. Write answers i...
 8.4.46: Find all complex number solutions of each equation. Write answers i...
 8.4.47: Find all complex number solutions of each equation. Write answers i...
 8.4.48: Find all complex number solutions of each equation. Write answers i...
 8.4.49: Solve each problem. Solve the cubic equationx3 = 1 by writing it as...
 8.4.50: Solve each problem. Solve the cubic equationx3 =27 by writing it a...
 8.4.51: Solve each problem. Mandelbrot Set The fractal known as the Mandelb...
 8.4.52: Solve each problem. Basins of Attraction The fractal shown in the f...
 8.4.53: Solve each problem. The screens here illustrate how a pentagon can ...
 8.4.54: Solve each problem. Use the method of Exercise 53 to find the first...
 8.4.55: Use a calculator to find all solutions of each equation in rectangu...
 8.4.56: Use a calculator to find all solutions of each equation in rectangu...
 8.4.57: Use a calculator to find all solutions of each equation in rectangu...
 8.4.58: Use a calculator to find all solutions of each equation in rectangu...
 8.4.59: For individual or collaborative investigation (Exercises 5962) In e...
 8.4.60: For individual or collaborative investigation (Exercises 5962) In e...
 8.4.61: For individual or collaborative investigation (Exercises 5962) In e...
 8.4.62: For individual or collaborative investigation (Exercises 5962) In e...
Solutions for Chapter 8.4: De Moivres Theorem; Powers and Roots of Complex Numbers
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 8.4: De Moivres Theorem; Powers and Roots of Complex Numbers
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Since 62 problems in chapter 8.4: De Moivres Theorem; Powers and Roots of Complex Numbers have been answered, more than 19524 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.4: De Moivres Theorem; Powers and Roots of Complex Numbers includes 62 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.