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Solutions for Chapter 2.4: Right Triangle Trigonometry

Trigonometry | ISBN: 9780495108351 | Authors: Charles P McKeague

Full solutions for Trigonometry

ISBN: 9780495108351

Trigonometry | ISBN: 9780495108351 | Authors: Charles P McKeague

Solutions for Chapter 2.4: Right Triangle Trigonometry

Solutions for Chapter 2.4
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Textbook: Trigonometry
Edition:
Author: Charles P McKeague
ISBN: 9780495108351

Chapter 2.4: Right Triangle Trigonometry includes 44 full step-by-step solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: . This expansive textbook survival guide covers the following chapters and their solutions. Since 44 problems in chapter 2.4: Right Triangle Trigonometry have been answered, more than 29580 students have viewed full step-by-step solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780495108351.

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).

  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Fibonacci numbers

    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].

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • Fundamental Theorem.

    The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • 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.

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • Orthogonal subspaces.

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

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Pivot.

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

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • 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 one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Similar matrices A and B.

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

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