 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 9606 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.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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 p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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