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# Solutions for Chapter 4.5: Graphs of Sine and Cosine Functions

## Full solutions for Precalculus With Limits A Graphing Approach | 5th Edition

ISBN: 9780618851522

Solutions for Chapter 4.5: Graphs of Sine and Cosine Functions

Solutions for Chapter 4.5
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##### ISBN: 9780618851522

Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Since 97 problems in chapter 4.5: Graphs of Sine and Cosine Functions have been answered, more than 103260 students have viewed full step-by-step solutions from this chapter. Chapter 4.5: Graphs of Sine and Cosine Functions includes 97 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: Precalculus With Limits A Graphing Approach, edition: 5.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• 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).

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• 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.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• 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.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Length II x II.

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

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• 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.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Vector addition.

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