 6.5.6.5.1: The _______ of a complex number is the distance between the origin ...
 6.5.6.5.2: The _______ of a complex number is given by where r is the _______ ...
 6.5.6.5.3: _______ Theorem states that if is a complex number and n is a posit...
 6.5.6.5.4: The complex number is an _______ of the complex number if z un a bi...
 6.5.6.5.5: In Exercises 18, plot the complex number and find its absolute value.
 6.5.6.5.6: In Exercises 18, plot the complex number and find its absolute value.
 6.5.6.5.7: In Exercises 18, plot the complex number and find its absolute value.
 6.5.6.5.8: In Exercises 18, plot the complex number and find its absolute value.
 6.5.6.5.9: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.10: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.11: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.12: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.13: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.14: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.15: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.16: In Exercises 916, write the complex number in trigonometric form wi...
 6.5.6.5.17: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.18: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.19: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.20: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.21: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.22: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.23: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.24: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.25: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.26: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.27: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.28: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.29: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.30: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.31: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.32: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.33: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.34: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.35: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.36: In Exercises 1736, represent the complex number graphically, and fi...
 6.5.6.5.37: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.38: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.39: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.40: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.41: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.42: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.43: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.44: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.45: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.46: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.47: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.48: In Exercises 3748, represent the complex number graphically, and fi...
 6.5.6.5.49: In Exercises 4952, use a graphing utility to represent the complex ...
 6.5.6.5.50: In Exercises 4952, use a graphing utility to represent the complex ...
 6.5.6.5.51: In Exercises 4952, use a graphing utility to represent the complex ...
 6.5.6.5.52: In Exercises 4952, use a graphing utility to represent the complex ...
 6.5.6.5.53: In Exercises 53 and 54, represent the powers and graphically. Descr...
 6.5.6.5.54: In Exercises 53 and 54, represent the powers and graphically. Descr...
 6.5.6.5.55: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.56: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.57: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.58: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.59: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.60: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.61: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.62: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.63: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.64: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.65: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.66: In Exercises 5566, perform the operation and leave the result in tr...
 6.5.6.5.67: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.68: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.69: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.70: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.71: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.72: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.73: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.74: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.75: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.76: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.77: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.78: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.79: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.80: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.81: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.82: In Exercises 6782, (a) write the trigonometric forms of the complex...
 6.5.6.5.83: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.84: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.85: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.86: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.87: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.88: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.89: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.90: In Exercises 8390, sketch the graph of all complex numbers z satisf...
 6.5.6.5.91: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.92: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.93: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.94: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.95: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.96: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.97: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.98: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.99: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.100: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.101: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.102: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.103: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.104: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.105: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.106: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.107: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.108: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.109: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.110: In Exercises 91110, use DeMoivres Theorem to find the indicated pow...
 6.5.6.5.111: Show that is a sixth root of 1.
 6.5.6.5.112: Show that is a fourth root of 2
 6.5.6.5.113: Graphical Reasoning In Exercises 113116, use the graph of the roots...
 6.5.6.5.114: Graphical Reasoning In Exercises 113116, use the graph of the roots...
 6.5.6.5.115: Graphical Reasoning In Exercises 113116, use the graph of the roots...
 6.5.6.5.116: Graphical Reasoning In Exercises 113116, use the graph of the roots...
 6.5.6.5.117: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.118: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.119: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.120: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.121: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.122: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.123: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.124: In Exercises 117124, find the square roots of the complex number.
 6.5.6.5.125: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.126: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.127: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.128: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.129: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.130: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.131: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.132: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.133: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.134: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.135: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.136: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.137: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.138: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.139: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.140: In Exercises 125140, (a) use the theorem on page 454 to find the in...
 6.5.6.5.141: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.142: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.143: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.144: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.145: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.146: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.147: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.148: In Exercises 141148, use the theorem on page 454 to find all the so...
 6.5.6.5.149: Electrical Engineering In Exercises 149154, use the formula to find...
 6.5.6.5.150: Electrical Engineering In Exercises 149154, use the formula to find...
 6.5.6.5.151: Electrical Engineering In Exercises 149154, use the formula to find...
 6.5.6.5.152: Electrical Engineering In Exercises 149154, use the formula to find...
 6.5.6.5.153: Electrical Engineering In Exercises 149154, use the formula to find...
 6.5.6.5.154: Electrical Engineering In Exercises 149154, use the formula to find...
 6.5.6.5.155: True or False? In Exercises 155157, determine whether the statement...
 6.5.6.5.156: True or False? In Exercises 155157, determine whether the statement...
 6.5.6.5.157: True or False? In Exercises 155157, determine whether the statement...
 6.5.6.5.158: Given two complex numbers and show that z1z 2 r1r2cos1 2 i sin1 2.z
 6.5.6.5.159: Show that is the complex conjugate of z rcos i sin .z
 6.5.6.5.160: Use the trigonometric forms of and in Exercise 159 to find (a) and ...
 6.5.6.5.161: Show that the negative of is z rcos i sin .z
 6.5.6.5.162: Writing The famous formula is called Eulers Formula, after the Swis...
 6.5.6.5.163: Harmonic Motion In Exercises 163166, for the simple harmonic motion...
 6.5.6.5.164: Harmonic Motion In Exercises 163166, for the simple harmonic motion...
 6.5.6.5.165: Harmonic Motion In Exercises 163166, for the simple harmonic motion...
 6.5.6.5.166: Harmonic Motion In Exercises 163166, for the simple harmonic motion...
 6.5.6.5.167: In Exercises 167170, find all solutions of the equation in the inte...
 6.5.6.5.168: In Exercises 167170, find all solutions of the equation in the inte...
 6.5.6.5.169: In Exercises 167170, find all solutions of the equation in the inte...
 6.5.6.5.170: In Exercises 167170, find all solutions of the equation in the inte...
Solutions for Chapter 6.5: Trigonometric Form of a Complex Number
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 6.5: Trigonometric Form of a Complex Number
Get Full SolutionsSince 170 problems in chapter 6.5: Trigonometric Form of a Complex Number have been answered, more than 33079 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 6.5: Trigonometric Form of a Complex Number includes 170 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

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·