 7.4.1: For an angle that lies in quadrant III, the trigonometric functions...
 7.4.2: Two angles in standard position that have the same terminal side are
 7.4.3: The reference angle of 240 is
 7.4.4: True or False sin 182 = cos 2
 7.4.5: True or False tan p2 is not defined.
 7.4.6: True or False The reference angle is always an acute angle.
 7.4.7: What is the reference angle of 600?
 7.4.8: In which quadrants is the cosine function positive?
 7.4.9: If 0 u 6 2p, for what angles u, if any, is tan u undefined?
 7.4.10: What is the reference angle of 13p3 ?
 7.4.11: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.12: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.13: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.14: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.15: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.16: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.17: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.18: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.19: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.20: In 1120, a point on the terminal side of an angle u in standard pos...
 7.4.21: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.22: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.23: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.24: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.25: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.26: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.27: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.28: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.29: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.30: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.31: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.32: In 2132, use a coterminal angle to find the exact value of each exp...
 7.4.33: In 3340, name the quadrant in which the angle u lies. sin u 7 0, co...
 7.4.34: In 3340, name the quadrant in which the angle u lies. sin u 6 0, co...
 7.4.35: In 3340, name the quadrant in which the angle u lies. sin u 6 0, ta...
 7.4.36: In 3340, name the quadrant in which the angle u lies. cos u 7 0, ta...
 7.4.37: In 3340, name the quadrant in which the angle u lies. cos u 7 0, co...
 7.4.38: In 3340, name the quadrant in which the angle u lies. sin u 6 0, co...
 7.4.39: In 3340, name the quadrant in which the angle u lies.sec u 6 0, tan...
 7.4.40: In 3340, name the quadrant in which the angle u lies.csc u 7 0, cot...
 7.4.41: In 4158, find the reference angle of each angle. 30
 7.4.42: In 4158, find the reference angle of each angle. 60
 7.4.43: In 4158, find the reference angle of each angle. 120
 7.4.44: In 4158, find the reference angle of each angle. 210
 7.4.45: In 4158, find the reference angle of each angle. 300
 7.4.46: In 4158, find the reference angle of each angle. 330
 7.4.47: In 4158, find the reference angle of each angle.5p 4
 7.4.48: In 4158, find the reference angle of each angle.5p 6
 7.4.49: In 4158, find the reference angle of each angle.8p 3
 7.4.50: In 4158, find the reference angle of each angle.7p 4
 7.4.51: In 4158, find the reference angle of each angle.135
 7.4.52: In 4158, find the reference angle of each angle.240
 7.4.53: In 4158, find the reference angle of each angle. 2p 3
 7.4.54: In 4158, find the reference angle of each angle. 7p 6
 7.4.55: In 4158, find the reference angle of each angle.440
 7.4.56: In 4158, find the reference angle of each angle.490
 7.4.57: In 4158, find the reference angle of each angle.15p 4
 7.4.58: In 4158, find the reference angle of each angle.19p 6
 7.4.59: In 5982, use the reference angle to find the exact value of each ex...
 7.4.60: In 5982, use the reference angle to find the exact value of each ex...
 7.4.61: In 5982, use the reference angle to find the exact value of each ex...
 7.4.62: In 5982, use the reference angle to find the exact value of each ex...
 7.4.63: In 5982, use the reference angle to find the exact value of each ex...
 7.4.64: In 5982, use the reference angle to find the exact value of each ex...
 7.4.65: In 5982, use the reference angle to find the exact value of each ex...
 7.4.66: In 5982, use the reference angle to find the exact value of each ex...
 7.4.67: In 5982, use the reference angle to find the exact value of each ex...
 7.4.68: In 5982, use the reference angle to find the exact value of each ex...
 7.4.69: In 5982, use the reference angle to find the exact value of each ex...
 7.4.70: In 5982, use the reference angle to find the exact value of each ex...
 7.4.71: In 5982, use the reference angle to find the exact value of each ex...
 7.4.72: In 5982, use the reference angle to find the exact value of each ex...
 7.4.73: In 5982, use the reference angle to find the exact value of each ex...
 7.4.74: In 5982, use the reference angle to find the exact value of each ex...
 7.4.75: In 5982, use the reference angle to find the exact value of each ex...
 7.4.76: In 5982, use the reference angle to find the exact value of each ex...
 7.4.77: In 5982, use the reference angle to find the exact value of each ex...
 7.4.78: In 5982, use the reference angle to find the exact value of each ex...
 7.4.79: In 5982, use the reference angle to find the exact value of each ex...
 7.4.80: In 5982, use the reference angle to find the exact value of each ex...
 7.4.81: In 5982, use the reference angle to find the exact value of each ex...
 7.4.82: In 5982, use the reference angle to find the exact value of each ex...
 7.4.83: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.84: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.85: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.86: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.87: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.88: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.89: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.90: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.91: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.92: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.93: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.94: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.95: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.96: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.97: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.98: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.99: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.100: In 83100, find the exact value of each of the remaining trigonometr...
 7.4.101: Find the exact value ofsin 40 + sin 130 + sin 220 + sin 310
 7.4.102: Find the exact value of tan40 + tan 140
 7.4.103: In 103106, f1x2 = sin x, g1x2 = cos x, h1x2 = tan x, F1x2 = csc x, ...
 7.4.104: In 103106, f1x2 = sin x, g1x2 = cos x, h1x2 = tan x, F1x2 = csc x, ...
 7.4.105: In 103106, f1x2 = sin x, g1x2 = cos x, h1x2 = tan x, F1x2 = csc x, ...
 7.4.106: In 103106, f1x2 = sin x, g1x2 = cos x, h1x2 = tan x, F1x2 = csc x, ...
 7.4.107: If f1u2 = sin u = 0.2 f1u + p2
 7.4.108: If g1u2 = cos u = 0.4 g1u + p2
 7.4.109: If F1u2 = tan u = 3 F1u + p2
 7.4.110: If G1u2 = cot u = 2 G1u + p
 7.4.111: If sin u = 15 csc1u + p2
 7.4.112: If cos u = 23 sec1u + p2
 7.4.113: Find the exact value ofsin 1 + sin 2 + sin 3 + + sin 358 + sin 359
 7.4.114: Find the exact value ofcos 1 + cos 2 + cos 3 + + cos 358 + cos 359
 7.4.115: Projectile Motion An object is propelled upward at an angle , to th...
 7.4.116: Give three examples that demonstrate how to use the theorem on refe...
 7.4.117: Write a brief paragraph that explains how to quickly compute the va...
 7.4.118: Explain what a reference angle is. What role does it play in findin...
Solutions for Chapter 7.4: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 7.4
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Since 118 problems in chapter 7.4 have been answered, more than 58247 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. Chapter 7.4 includes 118 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

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.

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.

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.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!