 6.5.1: In Exercises 16, fill in the blank to complete the trigonometric fo...
 6.5.2: In Exercises 16, fill in the blank to complete the trigonometric fo...
 6.5.3: In Exercises 16, fill in the blank to complete the trigonometric fo...
 6.5.4: In Exercises 16, fill in the blank to complete the trigonometric fo...
 6.5.5: In Exercises 16, fill in the blank to complete the trigonometric fo...
 6.5.6: In Exercises 16, fill in the blank to complete the trigonometric fo...
 6.5.7: Match each function with its doubleangle formula. (a) sin 2u (i) 1...
 6.5.8: Match each expression with its producttosum formula. (a) sin 2u (...
 6.5.9: In Exercises 9 and 10, use the figure to find the exact value of ea...
 6.5.10: In Exercises 9 and 10, use the figure to find the exact value of ea...
 6.5.11: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.12: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.13: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.14: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.15: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.16: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.17: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.18: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.19: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.20: In Exercises 1120, use a graphing utility to approximate the soluti...
 6.5.21: In Exercises 2126, find the exact values of sin 2u, cos 2u, and tan...
 6.5.22: In Exercises 2126, find the exact values of sin 2u, cos 2u, and tan...
 6.5.23: In Exercises 2126, find the exact values of sin 2u, cos 2u, and tan...
 6.5.24: In Exercises 2126, find the exact values of sin 2u, cos 2u, and tan...
 6.5.25: In Exercises 2126, find the exact values of sin 2u, cos 2u, and tan...
 6.5.26: In Exercises 2126, find the exact values of sin 2u, cos 2u, and tan...
 6.5.27: In Exercises 2730, use a doubleangle formula to rewrite the expres...
 6.5.28: In Exercises 2730, use a doubleangle formula to rewrite the expres...
 6.5.29: In Exercises 2730, use a doubleangle formula to rewrite the expres...
 6.5.30: In Exercises 2730, use a doubleangle formula to rewrite the expres...
 6.5.31: In Exercises 3134, rewrite the function using the powerreducing fo...
 6.5.32: In Exercises 3134, rewrite the function using the powerreducing fo...
 6.5.33: In Exercises 3134, rewrite the function using the powerreducing fo...
 6.5.34: In Exercises 3134, rewrite the function using the powerreducing fo...
 6.5.35: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.36: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.37: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.38: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.39: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.40: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.41: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.42: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.43: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.44: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.45: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.46: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.47: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.48: In Exercises 3548, rewrite the expression in terms of the first pow...
 6.5.49: In Exercises 49 and 50, use the figure to find the exact value of e...
 6.5.50: In Exercises 49 and 50, use the figure to find the exact value of e...
 6.5.51: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.52: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.53: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.54: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.55: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.56: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.57: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.58: In Exercises 5158, use the halfangle formulas to determine the exa...
 6.5.59: In Exercises 5964, find the exact values of sin(u2), cos(u2), and t...
 6.5.60: In Exercises 5964, find the exact values of sin(u2), cos(u2), and t...
 6.5.61: In Exercises 5964, find the exact values of sin(u2), cos(u2), and t...
 6.5.62: In Exercises 5964, find the exact values of sin(u2), cos(u2), and t...
 6.5.63: In Exercises 5964, find the exact values of sin(u2), cos(u2), and t...
 6.5.64: In Exercises 5964, find the exact values of sin(u2), cos(u2), and t...
 6.5.65: In Exercises 6568, use the halfangle formulas to simplify the expr...
 6.5.66: In Exercises 6568, use the halfangle formulas to simplify the expr...
 6.5.67: In Exercises 6568, use the halfangle formulas to simplify the expr...
 6.5.68: In Exercises 6568, use the halfangle formulas to simplify the expr...
 6.5.69: In Exercises 6972, find the solutions of the equation in the interv...
 6.5.70: In Exercises 6972, find the solutions of the equation in the interv...
 6.5.71: In Exercises 6972, find the solutions of the equation in the interv...
 6.5.72: In Exercises 6972, find the solutions of the equation in the interv...
 6.5.73: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.74: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.75: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.76: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.77: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.78: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.79: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.80: In Exercises 7380, use the producttosum formulas to write the pro...
 6.5.81: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.82: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.83: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.84: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.85: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.86: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.87: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.88: In Exercises 8188, use the sumtoproduct formulas to write the sum...
 6.5.89: In Exercises 8992, use the sumtoproduct formulas to find the exac...
 6.5.90: In Exercises 8992, use the sumtoproduct formulas to find the exac...
 6.5.91: In Exercises 8992, use the sumtoproduct formulas to find the exac...
 6.5.92: In Exercises 8992, use the sumtoproduct formulas to find the exac...
 6.5.93: In Exercises 9396, find the solutions of the equation in the interv...
 6.5.94: In Exercises 9396, find the solutions of the equation in the interv...
 6.5.95: In Exercises 9396, find the solutions of the equation in the interv...
 6.5.96: In Exercises 9396, find the solutions of the equation in the interv...
 6.5.97: In Exercises 97100, use the figure and trigonometric identities to ...
 6.5.98: In Exercises 97100, use the figure and trigonometric identities to ...
 6.5.99: In Exercises 97100, use the figure and trigonometric identities to ...
 6.5.100: In Exercises 97100, use the figure and trigonometric identities to ...
 6.5.101: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.102: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.103: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.104: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.105: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.106: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.107: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.108: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.109: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.110: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.111: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.112: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.113: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.114: In Exercises 101114, verify the identity algebraically. Use a graph...
 6.5.115: In Exercises 115120, write the trigonometric expression as an algeb...
 6.5.116: In Exercises 115120, write the trigonometric expression as an algeb...
 6.5.117: In Exercises 115120, write the trigonometric expression as an algeb...
 6.5.118: In Exercises 115120, write the trigonometric expression as an algeb...
 6.5.119: In Exercises 115120, write the trigonometric expression as an algeb...
 6.5.120: In Exercises 115120, write the trigonometric expression as an algeb...
 6.5.121: In Exercises 121 and 122, the graph of a function f is shown over t...
 6.5.122: In Exercises 121 and 122, the graph of a function f is shown over t...
 6.5.123: The range of a projectile fired at an angle with the horizontal and...
 6.5.124: The length of each of the two equal sides of an isosceles triangle ...
 6.5.125: When two railroad tracks merge, the overlapping portions of the tra...
 6.5.126: The Mach number M of an airplane is the ratio of its speed to the s...
 6.5.127: In Exercises 127 and 128, determine whether the statement is true o...
 6.5.128: In Exercises 127 and 128, determine whether the statement is true o...
 6.5.129: Consider the function f(x) = sin4 x + cos4 x. (a) Use the powerred...
 6.5.130: Explain how to use the figure to verify each doubleangle formula. ...
 6.5.131: In Exercises 131134, (a) plot the points, (b) find the distance bet...
 6.5.132: In Exercises 131134, (a) plot the points, (b) find the distance bet...
 6.5.133: In Exercises 131134, (a) plot the points, (b) find the distance bet...
 6.5.134: In Exercises 131134, (a) plot the points, (b) find the distance bet...
 6.5.135: In Exercises 135138, find (if possible) the complement and suppleme...
 6.5.136: In Exercises 135138, find (if possible) the complement and suppleme...
 6.5.137: In Exercises 135138, find (if possible) the complement and suppleme...
 6.5.138: In Exercises 135138, find (if possible) the complement and suppleme...
 6.5.139: In Exercises 139 and 140, find the length of the arc on a circle of...
 6.5.140: In Exercises 139 and 140, find the length of the arc on a circle of...
Solutions for Chapter 6.5: Analytic Trigonometry
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 6.5: Analytic Trigonometry
Get Full SolutionsSince 140 problems in chapter 6.5: Analytic Trigonometry have been answered, more than 60933 students have viewed full stepbystep solutions from this chapter. Chapter 6.5: Analytic Trigonometry includes 140 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.