 8.5.1: The distance d from the point 12, 32 to the point 15, 12 is
 8.5.2: If and is in quadrant II, then . sin u = u cos u = 4 5
 8.5.3: (a)sin p 4 # cos p 3 =(b) tan p 4  sin p 6 =
 8.5.4: If then . sin a =  cos a = 4 5 , p 6 a 6 3p 2 ,
 8.5.5: cos1a + b2 = cos a cos b sin a sin b
 8.5.6: sin1a  b2 = sin a cos b cos a sin b
 8.5.7: True or Falsesin1a + b2 = sin a + sin b + 2 sin a sin b
 8.5.8: True or Falsetan 75 = tan 30 + tan 45
 8.5.9: True or Falsecosa p 2  ub = cos u
 8.5.10: True or FalseThen g1a + b2 = g1a2g1b2  f1a2f1b2 f1x2 = sin x and g...
 8.5.11: In 1122, find the exact value of each expression.
 8.5.12: In 1122, find the exact value of each expression.
 8.5.13: In 1122, find the exact value of each expression.
 8.5.14: In 1122, find the exact value of each expression.
 8.5.15: In 1122, find the exact value of each expression.
 8.5.16: In 1122, find the exact value of each expression.
 8.5.17: In 1122, find the exact value of each expression.
 8.5.18: In 1122, find the exact value of each expression.
 8.5.19: In 1122, find the exact value of each expression.
 8.5.20: In 1122, find the exact value of each expression.
 8.5.21: In 1122, find the exact value of each expression.
 8.5.22: In 1122, find the exact value of each expression.
 8.5.23: In 2332, find the exact value of each expression
 8.5.24: In 2332, find the exact value of each expression
 8.5.25: In 2332, find the exact value of each expression
 8.5.26: In 2332, find the exact value of each expression
 8.5.27: In 2332, find the exact value of each expression
 8.5.28: In 2332, find the exact value of each expression
 8.5.29: In 2332, find the exact value of each expression
 8.5.30: In 2332, find the exact value of each expression
 8.5.31: In 2332, find the exact value of each expression
 8.5.32: In 2332, find the exact value of each expression
 8.5.33: In 3338, find the exact value of each of the following under the gi...
 8.5.34: In 3338, find the exact value of each of the following under the gi...
 8.5.35: In 3338, find the exact value of each of the following under the gi...
 8.5.36: In 3338, find the exact value of each of the following under the gi...
 8.5.37: In 3338, find the exact value of each of the following under the gi...
 8.5.38: In 3338, find the exact value of each of the following under the gi...
 8.5.39: If in quadrant II, find the exact value of: (a) (b) (c) (d) tanau +...
 8.5.40: If in quadrant IV, find the exact value of: (a) (b) (c) (d) tanau ...
 8.5.41: In 4146, use the figures to evaluate each function if f(x) = sin x,...
 8.5.42: In 4146, use the figures to evaluate each function if f(x) = sin x,...
 8.5.43: In 4146, use the figures to evaluate each function if f(x) = sin x,...
 8.5.44: In 4146, use the figures to evaluate each function if f(x) = sin x,...
 8.5.45: In 4146, use the figures to evaluate each function if f(x) = sin x,...
 8.5.46: In 4146, use the figures to evaluate each function if f(x) = sin x,...
 8.5.47: In 4772, establish each identity.
 8.5.48: In 4772, establish each identity.
 8.5.49: In 4772, establish each identity.
 8.5.50: In 4772, establish each identity.
 8.5.51: In 4772, establish each identity.
 8.5.52: In 4772, establish each identity.
 8.5.53: In 4772, establish each identity.
 8.5.54: In 4772, establish each identity.
 8.5.55: In 4772, establish each identity.
 8.5.56: In 4772, establish each identity.
 8.5.57: In 4772, establish each identity.
 8.5.58: In 4772, establish each identity.
 8.5.59: In 4772, establish each identity.
 8.5.60: In 4772, establish each identity.
 8.5.61: In 4772, establish each identity.
 8.5.62: In 4772, establish each identity.
 8.5.63: In 4772, establish each identity.
 8.5.64: In 4772, establish each identity.
 8.5.65: In 4772, establish each identity.
 8.5.66: In 4772, establish each identity.
 8.5.67: In 4772, establish each identity.
 8.5.68: In 4772, establish each identity.
 8.5.69: In 4772, establish each identity.
 8.5.70: In 4772, establish each identity.
 8.5.71: In 4772, establish each identity.
 8.5.72: In 4772, establish each identity.
 8.5.73: In 7384, find the exact value of each expression.
 8.5.74: In 7384, find the exact value of each expression.
 8.5.75: In 7384, find the exact value of each expression.
 8.5.76: In 7384, find the exact value of each expression.
 8.5.77: In 7384, find the exact value of each expression.
 8.5.78: In 7384, find the exact value of each expression.
 8.5.79: In 7384, find the exact value of each expression.
 8.5.80: In 7384, find the exact value of each expression.
 8.5.81: In 7384, find the exact value of each expression.
 8.5.82: In 7384, find the exact value of each expression.
 8.5.83: In 7384, find the exact value of each expression.
 8.5.84: In 7384, find the exact value of each expression.
 8.5.85: In 8590, write each trigonometric expression as an algebraic expres...
 8.5.86: In 8590, write each trigonometric expression as an algebraic expres...
 8.5.87: In 8590, write each trigonometric expression as an algebraic expres...
 8.5.88: In 8590, write each trigonometric expression as an algebraic expres...
 8.5.89: In 8590, write each trigonometric expression as an algebraic expres...
 8.5.90: In 8590, write each trigonometric expression as an algebraic expres...
 8.5.91: In 9196,solve each equation on the interval 0 u 6 2p.
 8.5.92: In 9196,solve each equation on the interval 0 u 6 2p.
 8.5.93: In 9196,solve each equation on the interval 0 u 6 2p.
 8.5.94: In 9196,solve each equation on the interval 0 u 6 2p.
 8.5.95: In 9196,solve each equation on the interval 0 u 6 2p.
 8.5.96: In 9196,solve each equation on the interval 0 u 6 2p.
 8.5.97: Show that sin1 v + cos1 v = p 2 .
 8.5.98: Show that tan1 v + cot1 v = p 2 .
 8.5.99: Show that tan if v 7 0. 1 a 1 v b = p 2  tan1 v,
 8.5.100: Show that cot1 ev = tan1 ev .
 8.5.101: Show that sin1sin1 v + cos1 v2 = 1.
 8.5.102: Show that cos1sin1 v + cos1 v2 = 0
 8.5.103: Calculus Show that the difference quotient for is given byf1x + h2 ...
 8.5.104: Calculus Show that the difference quotient for is given byh f1x + h...
 8.5.105: One, Two, Three (a) Show that (b) Conclude from part (a) that Sourc...
 8.5.106: Electric Power In an alternating current (ac) circuit, the instanta...
 8.5.107: Geometry: Angle between Two Lines Let and denote two nonvertical in...
 8.5.108: If sin3 u = sin1a  u2 sin1b  u2 sin1g  u2 cot u = cot a + cot b ...
 8.5.109: If and show that 2 cot1a  b2 = x2 tan a = x + 1 tan b = x  1
 8.5.110: Discuss the following derivation: Can you justify each step? = 1 t...
 8.5.111: Explain why formula (7) cannot be used to show that Establish this ...
Solutions for Chapter 8.5: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 8.5
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. This expansive textbook survival guide covers the following chapters and their solutions. Since 111 problems in chapter 8.5 have been answered, more than 55308 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Chapter 8.5 includes 111 full stepbystep solutions.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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 .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.