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Solutions for Chapter 4.4: Graphs of the Secant and Cosecant Functions

Trigonometry | 10th Edition | ISBN: 9780321671776 | Authors: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Full solutions for Trigonometry | 10th Edition

ISBN: 9780321671776

Trigonometry | 10th Edition | ISBN: 9780321671776 | Authors: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Solutions for Chapter 4.4: Graphs of the Secant and Cosecant Functions

Solutions for Chapter 4.4
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Textbook: Trigonometry
Edition: 10
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
ISBN: 9780321671776

Trigonometry was written by and is associated to the ISBN: 9780321671776. This expansive textbook survival guide covers the following chapters and their solutions. Since 36 problems in chapter 4.4: Graphs of the Secant and Cosecant Functions have been answered, more than 34275 students have viewed full step-by-step solutions from this chapter. Chapter 4.4: Graphs of the Secant and Cosecant Functions includes 36 full step-by-step solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 10.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Complete solution x = x p + Xn to Ax = b.

    (Particular x p) + (x n in nullspace).

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • 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.

  • 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 •

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • 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.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • 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!

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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