- Chapter 1: Trigonometric Functions
- Chapter 1.1: Angles
- Chapter 1.2: Angle Relationships and Similar Triangles
- Chapter 1.3: Trigonometric Functions
- Chapter 1.4: Using the Definitions of the Trigonometric Functions
- Chapter 2: Acute Angles and Right Triangles
- Chapter 2.1: Trigonometric Functions of Acute Angles
- Chapter 2.2: Trigonometric Functions of Non-Acute Angles
- Chapter 2.3: Approximations of Trigonometric Function Values
- Chapter 2.4: Solutions and Applications of Right Triangles
- Chapter 2.5: Further Applications of Right Triangles
- Chapter 3: Radian Measure and the Unit Circle
- Chapter 3.1: Radian Measure
- 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: Graphs of the Circular Functions
- Chapter 4.1: Graphs of the Sine and Cosine Functions
- 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: Trigonometric Identities
- Chapter 5.1: Fundamental Identities
- 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: Inverse Circular Functions and Trigonometric Equations
- Chapter 6.1: Inverse Circular Functions
- Chapter 6.2: Trigonometric Equations I
- Chapter 6.3: Trigonometric Equations II
- Chapter 6.4: Equations Involving Inverse Trigonometric Functions
- Chapter 7.1: Oblique Triangles and the Law of Sines
- Chapter 7.2: The Ambiguous Case of the Law of Sines
- Chapter 7.3: The Law of Cosines
- Chapter 7.4: Geometrically Defined Vectors and Applications
- Chapter 7.5: Algebraically Defined Vectors and the Dot Product
- Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations
- Chapter 8.1: Complex Numbers
- 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
Trigonometry 11th Edition - Solutions by Chapter
Full solutions for Trigonometry | 11th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
Column space C (A) =
space of all combinations of the columns of A.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
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 •
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.