 136.1: P (_5 13, _12 13) 2
 136.2: P (_ 2 2 , _ 2 2 ) Fi
 136.3: sin 240 4
 136.4: cos _10 3
 136.5: Find the period of this function.
 136.6: Graph the height of the spring as a function of time.
 136.7: P( _3 5 , _4 5)
 136.8: P( _12 13,  _5 13)
 136.9: P(_8 17, _15 17)
 136.10: P(_ 3 2 ,  _1 2)
 136.11: P( _1 2 , _ 3 2 ) 12
 136.12: P(0.6, 0.8)
 136.13: sin 690
 136.14: cos 750
 136.15: cos 5
 136.16: sin (_14 6 )
 136.17: sin ( _3 2 )
 136.18: cos (225)
 136.19: O y 1 34 2 5 6 7 8 11 1 9 10 2 13
 136.20: O y x 3 6 1 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54
 136.21: O y 2 3 4 5 1 1
 136.22: O y x 2 4 6 8 10 12 14 16 18
 136.23: Find the period of this function.
 136.24: Graph the height of the fixed point on the string from its resting ...
 136.25: cos 60 __+ sin 30 4
 136.26: 3(sin 60)(cos 30)
 136.27: sin 30  sin 60
 136.28: 4 cos 330 __+ 2 sin 60 3
 136.29: 12(sin 150)(cos 150)
 136.30: (sin 30)2 + (cos 30)2 1
 136.31: GEOMETRY A regular hexagon is inscribed in a unit circle centered a...
 136.32: BIOLOGY In a certain area of forested land, the population of rabbi...
 136.33: What is the slope of OP ?
 136.34: Which of the six trigonometric functions is equal to the slope of O...
 136.35: What is the slope of any line perpendicular to OP ? _
 136.36: Which of the six trigonometric functions is equal to the slope of a...
 136.37: Find the slope of OP when 60.
 136.38: If = 60, find the slope of the line tangent to circle O at point P. _
 136.39: OPEN ENDED Give an example of a situation that could be described b...
 136.40: WHICH ONE DOESNT BELONG? Identify the expression that does not belo...
 136.41: CHALLENGE Determine the domain and range of the functions y = sin a...
 136.42: Writing in Math If the formula for the temperature T in degrees Fah...
 136.43: ACT/SAT If ABC is an equilateral triangle, what is the length of AD...
 136.44: REVIEW For which measure of is = _ 3 3 ? F 135 G 270 H 1080 J 1830
 136.45: 45 17 15
 136.46: A C 5
 136.47: a = 11 in., c = 5 in., B = 79 4
 136.48: b = 4 m, c = 7 m, A = 63
 136.49: BULBS The lifetimes of 10,000 light bulbs are normally distributed....
 136.50: 16, 4, 1, _1 4 , 5
 136.51: n = 1 13(0.625)n  1
 136.52: sin = 0.3420
 136.53: cos = 0.3420
 136.54: tan = 3.2709
Solutions for Chapter 136: Circular Functions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 136: Circular Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 136: Circular Functions includes 54 full stepbystep solutions. Since 54 problems in chapter 136: Circular Functions have been answered, more than 56714 students have viewed full stepbystep solutions from this chapter. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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 .

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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