- 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
A = CTC = (L.J]))(L.J]))T for positive definite A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Gram-Schmidt 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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
= Xl (column 1) + ... + xn(column n) = combination of columns.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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