 146.1: cos = _3 5 ; 0 < < 90
 146.2: cos = _2 3 ; 180 < < 270
 146.3: sin = _1 2 ; 0 < < 90
 146.4: sin = _3 4 ; 270 < < 360
 146.5: sin 195
 146.6: cos _19 12
 146.7: AVIATION When a jet travels at speeds greater than the speed of sou...
 146.8: cot x = _sin 2x 1  cos 2x
 146.9: cos2 2x + 4 sin2 x cos2 x = 1
 146.10: sin = _5 13; 90 < < 180
 146.11: cos = _1 5 ; 270 < < 360
 146.12: cos = _1 3 ; 180 < < 270
 146.13: sin = _3 5 ; 180 < < 270
 146.14: sin = _3 8 ; 270 < < 360
 146.15: cos = _1 4 ; 90 < < 180
 146.16: cos 165
 146.17: sin 22 _1 2
 146.18: cos 157 _1 2
 146.19: sin 345
 146.20: sin _7 8
 146.21: cos _7 12
 146.22: sin 2x = 2 cot x sin2 x
 146.23: 2 cos2 _x 2 = 1 + cos x
 146.24: sin4 x  cos4 x = 2 sin2 x 1
 146.25: sin2 x = _1 2 (1  cos 2x)
 146.26: tan2 _x 2 = _1  cos x 1 + cos x
 146.27: 1 sin x cos x  _cos x sin x = tan x
 146.28: The horizontal distance d it will travel can be determined using th...
 146.29: The maximum height h the object will reach can be determined using ...
 146.30: cos = _1 6 ; 0 < < 90
 146.31: cos = _12 13; 180 < < 270
 146.32: sin = _1 3 ; 270 < < 360
 146.33: sin = _1 4 ; 180 < < 270
 146.34: cos = _2 3 ; 0 < < 90
 146.35: sin = _2 5 ; 90 < < 180
 146.36: OPTICS If a glass prism has an apex angle of measure and an angle o...
 146.37: Write this expression in terms of a trigonometric function of L.
 146.38: Find the exact value of the expression if L = 60.
 146.39: REASONING Explain how to find cos _x 2 if x is in the third quadrant.
 146.40: REASONING Describe the conditions under which you would use each of...
 146.41: OPEN ENDED Find a counterexample to show that cos 2 = 2 cos is not ...
 146.42: Writing in Math Use the information on page 853 to explain how trig...
 146.43: ACT/SAT Find the exact value of cos 2 if sin = _ 5 3 and 180 < < 2...
 146.44: REVIEW Which of the following is equivalent to cos (cot2 __ + 1) cs...
 146.45: cos 15
 146.46: sin 15
 146.47: sin (135)
 146.48: cos 150
 146.49: sin 105
 146.50: cos (300)
 146.51: cot2  sin2 = cos2 csc2  sin2 __ sin2 csc2
 146.52: cos (cos + cot ) = cot cos (sin + 1)
 146.53: How many times as great was the 1960 Chile earthquake as the 1938 I...
 146.54: The largest aftershock of the 1964 Alaskan earthquake was 6.7 on th...
 146.55: a8 7a4 + 13
 146.56: 5n7 + 3n 3
 146.57: d6 + 2d3 + 10
 146.58: f(2)
 146.59: f(0)
 146.60: f(3)
 146.61: f(n)
 146.62: (x + 6)(x  5) = 0
 146.63: (x  1)(x + 1) = 0
 146.64: x(x + 2) = 0
 146.65: (2x  5)(x + 2) = 0
 146.66: (2x + 1)(2x  1) = 0
 146.67: x2(2x + 1) = 0
Solutions for Chapter 146: DoubleAngle and HalfAngle Formulas
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 146: DoubleAngle and HalfAngle Formulas
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 146: DoubleAngle and HalfAngle Formulas includes 67 full stepbystep solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions. Since 67 problems in chapter 146: DoubleAngle and HalfAngle Formulas have been answered, more than 60413 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.