 8.1: Write each number as the product of a real number and i. 29
 8.2: Write each number as the product of a real number and i. 212
 8.3: Solve each equation over the set of complex numbers. . x2 =81
 8.4: Solve each equation over the set of complex numbers. x12x + 32=4
 8.5: Perform each operation. Write answers in standard form. 11  i213 ...
 8.6: Perform each operation. Write answers in standard form.12  5i2+19 ...
 8.7: Perform each operation. Write answers in standard form. 16  5i2+12...
 8.8: Perform each operation. Write answers in standard form.14  2i216 ...
 8.9: Perform each operation. Write answers in standard form. 13 + 5i218 ...
 8.10: Perform each operation. Write answers in standard form. 14  i215 + 2
 8.11: Perform each operation. Write answers in standard form. 12 + 6i22
 8.12: Perform each operation. Write answers in standard form.16  3i22
 8.13: Perform each operation. Write answers in standard form.11  i23
 8.14: Perform each operation. Write answers in standard form. 12 + i23
 8.15: Perform each operation. Write answers in standard form.25  19i 5 + 3i
 8.16: Perform each operation. Write answers in standard form.2  5i 1 + i
 8.17: Perform each operation. Write answers in standard form. 2 + i 1  5i
 8.18: Perform each operation. Write answers in standard form.3 + 2i i
 8.19: Perform each operation. Write answers in standard form. i53
 8.20: Perform each operation. Write answers in standard form. i41
 8.21: Perform each operation. Write answers in rectangular form. 351cos 9...
 8.22: Perform each operation. Write answers in rectangular form. 33 cis 1...
 8.23: Perform each operation. Write answers in rectangular form. 21cos 60...
 8.24: Perform each operation. Write answers in rectangular form. 4 cis 27...
 8.25: Perform each operation. Write answers in rectangular form. A23 + iB
 8.26: Perform each operation. Write answers in rectangular form. 12  2i25
 8.27: Perform each operation. Write answers in rectangular form. 1cos 100...
 8.28: Concept Check The vector representing a real number will lie on the...
 8.29: Graph each complex number. 5i
 8.30: Graph each complex number. 4 + 2i
 8.31: Graph each complex number. 3  3i23
 8.32: Find the sum of 7 + 3i and 2 + i. Graph both complex numbers and t...
 8.33: Write each complex number in its alternative form, using a calculat...
 8.34: Write each complex number in its alternative form, using a calculat...
 8.35: Write each complex number in its alternative form, using a calculat...
 8.36: Write each complex number in its alternative form, using a calculat...
 8.37: Write each complex number in its alternative form, using a calculat...
 8.38: Write each complex number in its alternative form, using a calculat...
 8.39: Write each complex number in its alternative form, using a calculat...
 8.40: Write each complex number in its alternative form, using a calculat...
 8.41: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.42: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.43: Find all roots as indicated. Write answers in trigonometric form. t...
 8.44: Find all roots as indicated. Write answers in trigonometric form. t...
 8.45: Concept Check How many real sixth roots does 64 have?
 8.46: Concept Check How many real fifth roots does 32 have?
 8.47: Find all complex number solutions. Write answers in trigonometric f...
 8.48: Find all complex number solutions. Write answers in trigonometric f...
 8.49: Find all complex number solutions. Write answers in trigonometric f...
 8.50: Convert 15, 3152 to rectangular coordinates.
 8.51: Convert A1, 23 B to polar coordinates, with 0 u 6 360 and r 7 0.
 8.52: Concept Check Describe the graph of r = k for k 7 0.
 8.53: Identify and graph each polar equation for u in 30, 3602. r = 4 cos u
 8.54: Identify and graph each polar equation for u in 30, 3602. r =1 + cos u
 8.55: Identify and graph each polar equation for u in 30, 3602. r = 2 sin 4u
 8.56: Identify and graph each polar equation for u in 30, 3602. r = 2 2 c...
 8.57: Find an equivalent equation in rectangular coordinates. r =3 1 + cos u
 8.58: Find an equivalent equation in rectangular coordinates. r = sin u +...
 8.59: Find an equivalent equation in rectangular coordinates. r = 2
 8.60: Find an equivalent equation in polar coordinates. y = x
 8.61: Find an equivalent equation in polar coordinates. y = x2
 8.62: Find an equivalent equation in polar coordinates. x2 + y2 = 25
 8.63: Identify the geometric symmetry (A, B, or C) that each graph will p...
 8.64: Identify the geometric symmetry (A, B, or C) that each graph will p...
 8.65: Identify the geometric symmetry (A, B, or C) that each graph will p...
 8.66: Identify the geometric symmetry (A, B, or C) that each graph will p...
 8.67: Find a polar equation having the given graph.
 8.68: Find a polar equation having the given graph.
 8.69: Find a polar equation having the given graph.
 8.70: Find a polar equation having the given graph.
 8.71: Graph the plane curve defined by the parametric equations x = t + c...
 8.72: Show that the distance between 1r1, u12 and 1r2, u22 in polar coord...
 8.73: Find a rectangular equation for each plane curve with the given par...
 8.74: Find a rectangular equation for each plane curve with the given par...
 8.75: Find a rectangular equation for each plane curve with the given par...
 8.76: Find a rectangular equation for each plane curve with the given par...
 8.77: Find a rectangular equation for each plane curve with the given par...
 8.78: Give a pair of parametric equations whose graph is the circle havin...
 8.79: (Modeling) Flight of a Baseball A batter hits a baseball when it is...
 8.80: Mandelbrot Set Consider the complex number z = 1 + i. Compute the v...
Solutions for Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Since 80 problems in chapter 8: Complex Numbers, Polar Equations, and Parametric Equations have been answered, more than 10421 students have viewed full stepbystep solutions from this chapter. Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations includes 80 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·
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