- 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
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.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
peA) = det(A - AI) has peA) = zero matrix.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Outer product uv T
= column times row = rank one matrix.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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