Solutions for Chapter 5.3: Sum and Difference Identities for Cosine

Full solutions for Trigonometry | 11th Edition

ISBN: 9780134217437

Solutions for Chapter 5.3: Sum and Difference Identities for Cosine

Solutions for Chapter 5.3
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Textbook: Trigonometry
Edition: 11
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
ISBN: 9780134217437

Trigonometry was written by and is associated to the ISBN: 9780134217437. Since 82 problems in chapter 5.3: Sum and Difference Identities for Cosine have been answered, more than 9607 students have viewed full step-by-step solutions from this chapter. Chapter 5.3: Sum and Difference Identities for Cosine includes 82 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 11.

Key Math Terms and definitions covered in this textbook
  • 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).

  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • 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).

  • Conjugate Gradient Method.

    A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

  • Cyclic shift

    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.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Elimination.

    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.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Kirchhoff's Laws.

    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.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • 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).

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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