 125.1: William said that sin (A 1 B) 1 sin (A 2 B) 5 sin 2A. Do you agree ...
 125.2: Freddy said that sin (A 1 B) 1 sin (A 2 B) 5 2 sin A cos B. Do you ...
 125.3: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.4: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.5: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.6: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.7: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.8: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.9: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.10: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.11: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.12: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.13: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.14: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.15: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.16: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.17: In 317, find the exact value of sin (A 2 B) and of sin (A 1 B) for ...
 125.18: In 1820, show all work. a. Find the exact value of sin 15 by using ...
 125.19: In 1820, show all work. a. Find the exact value of sin 120 by using...
 125.20: In 1820, show all work. a. Find the exact value of sin 210 by using...
 125.21: A telephone pole is braced by two wires that are both fastened to t...
 125.22: Two boats leave the same dock to cross a river that is 500 feet wid...
 125.23: The coordinates of any point in the coordinate plane can be written...
Solutions for Chapter 125: SINE (A 2 B) AND SINE (A 1 B)
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 125: SINE (A 2 B) AND SINE (A 1 B)
Get Full SolutionsSince 23 problems in chapter 125: SINE (A 2 B) AND SINE (A 1 B) have been answered, more than 28438 students have viewed full stepbystep solutions from this chapter. Chapter 125: SINE (A 2 B) AND SINE (A 1 B) includes 23 full stepbystep solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
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 nl  An).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Solvable system Ax = b.
The right side b is in the column space of A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.