- Chapter 1: Chapter 1 Review Exercises
- Chapter 1.1: Angles
- Chapter 1.11.2: Quiz
- Chapter 1.2: Trigonometric Functions
- Chapter 1.3: Trigonometric Functions
- Chapter 1.4: Using the Definitions of the Trigonometric Functions
- Chapter 2: Chapter 2 Review Exercises
- Chapter 2.1: Trigonometric Functions of Acute Angles
- Chapter 2.1-2.3: Quiz
- Chapter 2.2: Trigonometric Functions of Non-Acute Angles
- Chapter 2.3: Finding Trigonometric Function Values Using a Calculator
- Chapter 2.4: Solving Right Triangles
- Chapter 2.5: Further Applications of Right Triangles
- Chapter 3: Review Exercises
- Chapter 3.1: Radian Measure
- Chapter 3.1 - 3.3: Quiz
- Chapter 3.2: Applications of Radian Measure
- Chapter 3.3: The Unit Circle and Circular Functions
- Chapter 3.4: Linear and Angular Speed
- Chapter 4: Review Exercises
- Chapter 4.1: Graphs of the Sine and Cosine Functions
- Chapter 4.1 - 4.2: Quiz
- Chapter 4.2: Translations of the Graphs of the Sine and Cosine Functions
- Chapter 4.3: Graphs of the Tangent and Cotangent Functions
- Chapter 4.4: Graphs of the Secant and Cosecant Functions
- Chapter 4.5: Harmonic Motion
- Chapter 5: Review Exercise
- Chapter 5.1: Fundamental Identities
- Chapter 5.1 - 5.4: Quiz
- Chapter 5.2: Verifying Trigonometric Identities
- Chapter 5.3: Sum and Difference Identities for Cosine
- Chapter 5.4: Sum and Difference Identities for Sine and Tangent
- Chapter 5.5: Double-Angle Identities
- Chapter 5.6: Half-Angle Identities
- Chapter 6: Review Exercises
- Chapter 6.1: Inverse Circular Functions
- Chapter 6.1 - 6.3: Quiz
- Chapter 6.2: Trigonometric Equations I
- Chapter 6.3: Trigonometric Equations II
- Chapter 6.4: Equations Involving Inverse Trigonometric Functions
- Chapter 7: Summary Exercises on Applications of Trigonometry and Vectors
- Chapter 7.1: Oblique Triangles and the Law of Sines
- Chapter 7.1 - 7.3: Quiz
- Chapter 7.2: The Ambiguous Case of the Law of Sines
- Chapter 7.3: The Law of Cosines
- Chapter 7.4: Vectors, Operations, and the Dot Product
- Chapter 7.5: Applications of Vectors
- Chapter 8: Review Exercises
- Chapter 8.1: Complex Numbers
- Chapter 8.1 - 8.4: Quiz
- Chapter 8.2: Trigonometric (Polar) Form of Complex Numbers
- Chapter 8.3: The Product and Quotient Theorems
- Chapter 8.4: De Moivres Theorem; Powers and Roots of Complex Numbers
- Chapter 8.5: Polar Equations and Graphs
- Chapter 8.6: Parametric Equations, Graphs, and Applications
- Chapter Appendix A: Appendix A Exercises
- Chapter Appendix B: Appendix B Exercises
- Chapter Appendix C: Appendix C Exercises
- Chapter Appendix D: Appendix D Exercises
Trigonometry 10th Edition - Solutions by Chapter
Full solutions for Trigonometry | 10th Edition
Column space C (A) =
space of all combinations of the columns of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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 .
lA-II = 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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.