 6.3.1: CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and...
 6.3.2: CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and...
 6.3.3: CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and...
 6.3.4: CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and...
 6.3.5: CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and...
 6.3.6: CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and...
 6.3.7: CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and...
 6.3.8: CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and...
 6.3.9: CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and...
 6.3.10: CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and...
 6.3.11: CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and...
 6.3.12: CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and...
 6.3.13: Suppose solving a trigonometric equation for solutions over the int...
 6.3.14: Suppose solving a trigonometric equation for solutions over the int...
 6.3.15: Suppose solving a trigonometric equation for solutions over the int...
 6.3.16: Suppose solving a trigonometric equation for solutions over the int...
 6.3.17: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.18: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.19: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.20: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.21: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.22: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.23: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.24: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.25: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.26: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.27: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.28: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.29: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.30: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.31: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.32: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.33: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.34: Solve each equation in x for exact solutions over the interval 30, ...
 6.3.35: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.36: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.37: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.38: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.39: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.40: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.41: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.42: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.43: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.44: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.45: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.46: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.47: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.48: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.49: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.50: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.51: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.52: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.53: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.54: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.55: The following equations cannot be solved by algebraic methods. Use ...
 6.3.56: The following equations cannot be solved by algebraic methods. Use ...
 6.3.57: Pressure of a Plucked String If a string with a fundamental frequen...
 6.3.58: Hearing Beats in Music Musicians sometimes tune instruments by play...
 6.3.59: Hearing Difference Tones When a musical instrument creates a tone o...
 6.3.60: Daylight Hours in New Orleans The seasonal variation in length of d...
 6.3.61: Average Monthly Temperature in Vancouver The following function app...
 6.3.62: Average Monthly Temperature in Phoenix The following function appro...
 6.3.63: (Modeling) Alternating Electric Current The study of alternating el...
 6.3.64: (Modeling) Alternating Electric Current The study of alternating el...
 6.3.65: (Modeling) Alternating Electric Current The study of alternating el...
 6.3.66: (Modeling) Alternating Electric Current The study of alternating el...
Solutions for Chapter 6.3: Trigonometric Equations II
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 6.3: Trigonometric Equations II
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9780134217437. Chapter 6.3: Trigonometric Equations II includes 66 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 11. Since 66 problems in chapter 6.3: Trigonometric Equations II have been answered, more than 24151 students have viewed full stepbystep solutions from this chapter.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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 .

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 •

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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