 7.5.1: In Exercises 13, fill in the blank. 1. The _______ of a complex num...
 7.5.2: In Exercises 13, fill in the blank. 1. The _______ of a complex num...
 7.5.3: In Exercises 13, fill in the blank. 1. The _______ of a complex num...
 7.5.4: What is the trigonometric form of the complex number z = a + bi?
 7.5.5: When a complex number is written in trigonometric form, what does r...
 7.5.6: When a complex number is written in trigonometric form, what does r...
 7.5.7: In Exercises 714, plot the complex number and find its absolute val...
 7.5.8: In Exercises 714, plot the complex number and find its absolute val...
 7.5.9: In Exercises 714, plot the complex number and find its absolute val...
 7.5.10: In Exercises 714, plot the complex number and find its absolute val...
 7.5.11: In Exercises 714, plot the complex number and find its absolute val...
 7.5.12: In Exercises 714, plot the complex number and find its absolute val...
 7.5.13: In Exercises 714, plot the complex number and find its absolute val...
 7.5.14: In Exercises 714, plot the complex number and find its absolute val...
 7.5.15: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.16: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.17: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.18: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.19: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.20: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.21: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.22: In Exercises 1522, write the complex number in trigonometric form w...
 7.5.23: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.24: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.25: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.26: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.27: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.28: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.29: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.30: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.31: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.32: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.33: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.34: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.35: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.36: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.37: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.38: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.39: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.40: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.41: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.42: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.43: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.44: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.45: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.46: In Exercises 23 46, represent the complex number graphically, and f...
 7.5.47: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.48: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.49: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.50: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.51: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.52: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.53: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.54: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.55: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.56: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.57: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.58: In Exercises 4758, represent the complex number graphically, and fi...
 7.5.59: In Exercises 5962, use a graphing utility to represent the complex ...
 7.5.60: In Exercises 5962, use a graphing utility to represent the complex ...
 7.5.61: In Exercises 5962, use a graphing utility to represent the complex ...
 7.5.62: In Exercises 5962, use a graphing utility to represent the complex ...
 7.5.63: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.64: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.65: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.66: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.67: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.68: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.69: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.70: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.71: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.72: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.73: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.74: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.75: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.76: In Exercises 6376, perform the operation and leave the result in tr...
 7.5.77: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.78: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.79: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.80: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.81: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.82: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.83: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.84: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.85: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.86: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.87: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.88: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.89: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.90: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.91: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.92: In Exercises 7792, (a) write the trigonometric forms of the complex...
 7.5.93: In Exercises 93 and 94, represent the powers z, z2, z3, and z4 grap...
 7.5.94: In Exercises 93 and 94, represent the powers z, z2, z3, and z4 grap...
 7.5.95: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.96: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.97: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.98: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.99: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.100: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.101: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.102: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.103: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.104: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.105: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.106: In Exercises 95106, sketch the graph of all complex numbers z satis...
 7.5.107: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.108: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.109: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.110: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.111: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.112: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.113: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.114: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.115: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.116: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.117: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.118: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.119: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.120: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.121: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.122: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.123: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.124: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.125: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.126: In Exercises 107126, use DeMoivres Theorem to find the indicated po...
 7.5.127: In Exercises 127 and 128, use DeMoivres Theorem to verify the indic...
 7.5.128: In Exercises 127 and 128, use DeMoivres Theorem to verify the indic...
 7.5.129: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.130: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.131: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.132: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.133: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.134: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.135: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.136: In Exercises 129136, find the square roots of the complex number. 1...
 7.5.137: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.138: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.139: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.140: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.141: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.142: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.143: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.144: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.145: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.146: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.147: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.148: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.149: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.150: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.151: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.152: In Exercises 137152, (a) use the theorem on page 590 to find the in...
 7.5.153: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.154: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.155: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.156: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.157: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.158: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.159: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.160: In Exercises 153160, use the theorem on page 590 to find all the so...
 7.5.161: The formula E = IZ, where E represents voltage, I represents curren...
 7.5.162: The formula E = IZ, where E represents voltage, I represents curren...
 7.5.163: The formula E = IZ, where E represents voltage, I represents curren...
 7.5.164: The formula E = IZ, where E represents voltage, I represents curren...
 7.5.165: The formula E = IZ, where E represents voltage, I represents curren...
 7.5.166: The formula E = IZ, where E represents voltage, I represents curren...
 7.5.167: In Exercises 167170, determine whether the statement is true or fal...
 7.5.168: In Exercises 167170, determine whether the statement is true or fal...
 7.5.169: In Exercises 167170, determine whether the statement is true or fal...
 7.5.170: In Exercises 167170, determine whether the statement is true or fal...
 7.5.171: Given two complex numbers z1 = r1(cos 1 + i sin 1) and z2 = r2(cos ...
 7.5.172: Show that z = r[cos() + i sin()] is the complex conjugate of z = r(...
 7.5.173: Use trigonometric forms of z and z in Exercise 172 to find the foll...
 7.5.174: Show that the negative of z = r(cos + i sin ) is z = r[cos( + ) + i...
 7.5.175: The famous formula ea+bi = ea(cos b + i sin b) is called Eulers For...
 7.5.176: One of the fourth roots of a complex number z is shown in the figur...
 7.5.177: In Exercises 177180, for the simple harmonic motion described by th...
 7.5.178: In Exercises 177180, for the simple harmonic motion described by th...
 7.5.179: In Exercises 177180, for the simple harmonic motion described by th...
 7.5.180: In Exercises 177180, for the simple harmonic motion described by th...
Solutions for Chapter 7.5: Additional Topics in Trigonometry
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 7.5: Additional Topics in Trigonometry
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Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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.

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.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.