- Chapter 14.1: sin 135
- Chapter 14.2: tan 315
- Chapter 14.3: cos 90
- Chapter 14.4: tan 45
- Chapter 14.5: sin _5 4
- Chapter 14.6: cos _7 6
- Chapter 14.7: cos (-150)
- Chapter 14.8: cot _9 4
- Chapter 14.9: sec _13 6
- Chapter 14.10: tan (- _3 2 ) 1
- Chapter 14.11: tan _8 3
- Chapter 14.12: csc (-720)
- Chapter 14.13: AMUSEMENT The distance from the highest point of a Ferris wheel to ...
- Chapter 14.14: -15x2 - 5x
- Chapter 14.15: 2x4 - 4x2
- Chapter 14.16: x3 +
- Chapter 14.17: 2x2 - 3x - 2
- Chapter 14.18: PARKS The rectangular wooded area of a park covers x2 - 6x + 8 squa...
- Chapter 14.19: x2 - 5x - 24 = 0
- Chapter 14.20: x2 - 2x - 48 = 0
- Chapter 14.21: x2 - 12x = 0
- Chapter 14.22: x2 - 16 = 0
- Chapter 14.23: HOME IMPROVEMENT You are putting new flooring in your laundry room,...
- Chapter 14.24: sin csc - cos2
- Chapter 14.25: cos2 sec csc
- Chapter 14.26: cos + sin tan
- Chapter 14.27: sin (1 + cot2 )
- Chapter 14.28: PHYSICS The magnetic force on a particle can be modeled by the equa...
- Chapter 14.29: sin tan + _cos cot = cos + sin
- Chapter 14.30: sin 1 - cos = csc + cot
- Chapter 14.31: cot2 sec2 = 1 + cot2
- Chapter 14.32: sec ( sec - cos ) = tan2
- Chapter 14.33: OPTICS The amount of light passing through a polarization filter ca...
- Chapter 14.34: cos 15
- Chapter 14.35: cos 285
- Chapter 14.36: sin 150
- Chapter 14.37: sin 195
- Chapter 14.38: cos (-210)
- Chapter 14.39: sin (-105)
- Chapter 14.40: cos (90 + ) = -sin
- Chapter 14.41: sin (30 ) = cos (60 + )
- Chapter 14.42: sin ( + ) = sin
- Chapter 14.43: cos = (cos + )
- Chapter 14.44: sin = _1 4 ; 0 < < 90
- Chapter 14.45: sin = _5 13; 180 < < 270
- Chapter 14.46: cos = _5 17 ; 90 < < 180
- Chapter 14.47: cos = _12 13; 270 < < 360
- Chapter 14.48: 2 sin 2 = 1
- Chapter 14.49: cos2 + sin2 = 2 cos 0
- Chapter 14.50: PRISMS The horizontal and vertical components of an oblique prism c...
Solutions for Chapter Chapter 14: Trigonometric Graphs and Identities
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
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.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).