 7.7.1: In 1 6, find the exact value of each expression.
 7.7.2: In 1 6, find the exact value of each expression.
 7.7.3: In 1 6, find the exact value of each expression.
 7.7.4: In 1 6, find the exact value of each expression.
 7.7.5: In 1 6, find the exact value of each expression.
 7.7.6: In 1 6, find the exact value of each expression.
 7.7.7: In 716, express each product as a sum containing only sines or only...
 7.7.8: In 716, express each product as a sum containing only sines or only...
 7.7.9: In 716, express each product as a sum containing only sines or only...
 7.7.10: In 716, express each product as a sum containing only sines or only...
 7.7.11: In 716, express each product as a sum containing only sines or only...
 7.7.12: In 716, express each product as a sum containing only sines or only...
 7.7.13: In 716, express each product as a sum containing only sines or only...
 7.7.14: In 716, express each product as a sum containing only sines or only...
 7.7.15: In 716, express each product as a sum containing only sines or only...
 7.7.16: In 716, express each product as a sum containing only sines or only...
 7.7.17: In 1724, express each sum or difference as a product of sines and/o...
 7.7.18: In 1724, express each sum or difference as a product of sines and/o...
 7.7.19: In 1724, express each sum or difference as a product of sines and/o...
 7.7.20: In 1724, express each sum or difference as a product of sines and/o...
 7.7.21: In 1724, express each sum or difference as a product of sines and/o...
 7.7.22: In 1724, express each sum or difference as a product of sines and/o...
 7.7.23: In 1724, express each sum or difference as a product of sines and/o...
 7.7.24: In 1724, express each sum or difference as a product of sines and/o...
 7.7.25: In 2542, establish each identity
 7.7.26: In 2542, establish each identity
 7.7.27: In 2542, establish each identity
 7.7.28: In 2542, establish each identity
 7.7.29: In 2542, establish each identity
 7.7.30: In 2542, establish each identity
 7.7.31: In 2542, establish each identity
 7.7.32: In 2542, establish each identity
 7.7.33: In 2542, establish each identity
 7.7.34: In 2542, establish each identity
 7.7.35: In 2542, establish each identity
 7.7.36: In 2542, establish each identity
 7.7.37: In 2542, establish each identity
 7.7.38: In 2542, establish each identity
 7.7.39: In 2542, establish each identity
 7.7.40: In 2542, establish each identity
 7.7.41: In 2542, establish each identity
 7.7.42: In 2542, establish each identity
 7.7.43: In 4346, solve each equation on the interval 0 u 6 2p
 7.7.44: In 4346, solve each equation on the interval 0 u 6 2p
 7.7.45: In 4346, solve each equation on the interval 0 u 6 2p
 7.7.46: In 4346, solve each equation on the interval 0 u 6 2p
 7.7.47: On a TouchTone phone, each button produces a unique sound. The sou...
 7.7.48: (a) Write the sound emitted by touching the # key as a product of s...
 7.7.49: The moment of inertia I of an object is a measure of how easy it is...
 7.7.50: The range R of a projectile propelled downward from the top of an i...
 7.7.51: If a + b + g = p, show that sin12a2 + sin12b2 + sin12g2 = 4 sin a s...
 7.7.52: If a + b + g = p, show that tan a + tan b + tan g = tan a tan b tan g
 7.7.53: Derive formula (3).
 7.7.54: Derive formula (7).
 7.7.55: Derive formula (8).
 7.7.56: Derive formula (9).
Solutions for Chapter 7.7: ProducttoSum and SumtoProduct Formulas
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 7.7: ProducttoSum and SumtoProduct Formulas
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Chapter 7.7: ProducttoSum and SumtoProduct Formulas includes 56 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 7.7: ProducttoSum and SumtoProduct Formulas have been answered, more than 54465 students have viewed full stepbystep solutions from this chapter.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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.

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.

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).