 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: Find each product and write it in rectangular form. See Example 1. ...
 8.3.4: Find each product and write it in rectangular form. See Example 1. ...
 8.3.5: Find each product and write it in rectangular form. See Example 1. ...
 8.3.6: Find each product and write it in rectangular form. See Example 1. ...
 8.3.7: Find each product and write it in rectangular form. See Example 1. ...
 8.3.8: Find each product and write it in rectangular form. See Example 1. ...
 8.3.9: Find each product and write it in rectangular form. See Example 1. ...
 8.3.10: Find each product and write it in rectangular form. See Example 1. ...
 8.3.11: Find each product and write it in rectangular form. See Example 1. ...
 8.3.12: Find each product and write it in rectangular form. See Example 1. ...
 8.3.13: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.14: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.15: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.16: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.17: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.18: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.19: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.20: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.21: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.22: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.23: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.24: Find each quotient and write it in rectangular form. In Exercises 1...
 8.3.25: Use a calculator to perform the indicated operations. Give answers ...
 8.3.26: Use a calculator to perform the indicated operations. Give answers ...
 8.3.27: Use a calculator to perform the indicated operations. Give answers ...
 8.3.28: Use a calculator to perform the indicated operations. Give answers ...
 8.3.29: Use a calculator to perform the indicated operations. Give answers ...
 8.3.30: Use a calculator to perform the indicated operations. Give answers ...
 8.3.31: Use a calculator to perform the indicated operations. Give answers ...
 8.3.32: Use a calculator to perform the indicated operations. Give answers ...
 8.3.33: Multiply w and z using their rectangular forms and the FOIL method ...
 8.3.34: Find the trigonometric forms of w and z.
 8.3.35: Multiply w and z using their trigonometric forms and the method des...
 8.3.36: Use the result of Exercise 35 to find the rectangular form of wz. H...
 8.3.37: Find the quotient wz using their rectangular forms and multiplying ...
 8.3.38: Use the trigonometric forms of w and z, found in Exercise 34, to di...
 8.3.39: Use the result of Exercise 38 to find the rectangular form of wz . ...
 8.3.40: Note that 1r cis u22 = 1r cis u21r cis u2 = r2 cis1u + u2 = r2 cis ...
 8.3.41: Without actually performing the operations, state why the following...
 8.3.42: Show that 1z = 1r 1cos u  i sin u2, where z = r1cos u + i sin u2.
 8.3.43: Electrical Current The alternating current in an electric inductor ...
 8.3.44: Electrical Current The current I in a circuit with voltage E, resis...
 8.3.45: If Z1 = 50 + 25i and Z2 = 60 + 20i, calculate Z.
 8.3.46: Determine the angle u for the value of Z found in Exercise 45.
Solutions for Chapter 8.3: The Product and Quotient Theorems
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 8.3: The Product and Quotient Theorems
Get Full SolutionsChapter 8.3: The Product and Quotient Theorems includes 46 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: 9780321671776. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Since 46 problems in chapter 8.3: The Product and Quotient Theorems have been answered, more than 33072 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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 ().

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

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.

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