- 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
Remove row i and column j; multiply the determinant by (-I)i + j •
Column space C (A) =
space of all combinations of the columns of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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].
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.
A sequence of steps intended to approach the desired solution.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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 •
Outer product uv T
= column times row = rank one matrix.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).