 8.3.1: When multiplying two complex numbers in trigonometric form, we thei...
 8.3.2: When dividing two complex numbers in trigonometric form, we their a...
 8.3.3: 351cos 150 + i sin 15024321cos 30 + i sin 3024 = 1cos + i sin 2 = + i
 8.3.4: 61cos 120 + i sin 1202 21cos 30 + i sin 302 = 1cos + i sin 2 = + i ...
 8.3.5: cis110002# cis 1000 = cis = + i
 8.3.6: 5 cis 50,000 cis 50,000 = 5 cis = + i
 8.3.7: Find each product. Write answers in rectangular form. See Example 1...
 8.3.8: Find each product. Write answers in rectangular form. See Example 1...
 8.3.9: Find each product. Write answers in rectangular form. See Example 1...
 8.3.10: Find each product. Write answers in rectangular form. See Example 1...
 8.3.11: Find each product. Write answers in rectangular form. See Example 1...
 8.3.12: Find each product. Write answers in rectangular form. See Example 1...
 8.3.13: Find each product. Write answers in rectangular form. See Example 1...
 8.3.14: Find each product. Write answers in rectangular form. See Example 1...
 8.3.15: Find each product. Write answers in rectangular form. See Example 1...
 8.3.16: Find each product. Write answers in rectangular form. See Example 1...
 8.3.17: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.18: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.19: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.20: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.21: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.22: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.23: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.24: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.25: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.26: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.27: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.28: Find each quotient. Write answers in rectangular form. In Exercises...
 8.3.29: Use a calculator to perform the indicated operations. Write answers...
 8.3.30: Use a calculator to perform the indicated operations. Write answers...
 8.3.31: Use a calculator to perform the indicated operations. Write answers...
 8.3.32: Use a calculator to perform the indicated operations. Write answers...
 8.3.33: Use a calculator to perform the indicated operations. Write answers...
 8.3.34: Use a calculator to perform the indicated operations. Write answers...
 8.3.35: Use a calculator to perform the indicated operations. Write answers...
 8.3.36: Use a calculator to perform the indicated operations. Write answers...
 8.3.37: Work each problem.Note that 1r cis u22 =1r cis u21r cis u2= r2 cis1...
 8.3.38: Work each problem.Without actually performing the operations, state...
 8.3.39: Work each problem.Show that 1 z = 1 r 1cos u  i sin u2, where z = ...
 8.3.40: Work each problem.The complex conjugate of r 1cos u + i sin u2 is r...
 8.3.41: (Modeling) Electrical Current Solve each problem. The alternating c...
 8.3.42: (Modeling) Electrical Current Solve each problem. The current I in ...
 8.3.43: (Modeling) Electrical Current Solve each problem. (Modeling) Impeda...
 8.3.44: (Modeling) Electrical Current Solve each problem. (Modeling) Impeda...
 8.3.45: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.46: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.47: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.48: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.49: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.50: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.51: Consider the following complex numbers, and work Exercises 4552 in ...
 8.3.52: Consider the following complex numbers, and work Exercises 4552 in ...
Solutions for Chapter 8.3: The Product and Quotient Theorems
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 8.3: The Product and Quotient Theorems
Get Full SolutionsSince 52 problems in chapter 8.3: The Product and Quotient Theorems have been answered, more than 9596 students have viewed full stepbystep solutions from this chapter. Chapter 8.3: The Product and Quotient Theorems includes 52 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. This textbook survival guide was created for the textbook: Trigonometry, edition: 11.

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).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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).

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

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 •

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 ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

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.

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