 5.5.5.5.1: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.2: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.3: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.4: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.5: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.6: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.7: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.8: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.9: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.10: Fill in the blank to complete the trigonometric formula.
 5.5.5.5.11: In Exercises 312, use a graphing utility to approximate the solutio...
 5.5.5.5.12: In Exercises 312, use a graphing utility to approximate the solutio...
 5.5.5.5.13: In Exercises 1318, find the exact values of and using the doublean...
 5.5.5.5.14: In Exercises 1318, find the exact values of and using the doublean...
 5.5.5.5.15: In Exercises 1318, find the exact values of and using the doublean...
 5.5.5.5.16: In Exercises 1318, find the exact values of and using the doublean...
 5.5.5.5.17: In Exercises 1318, find the exact values of and using the doublean...
 5.5.5.5.18: In Exercises 1318, find the exact values of and using the doublean...
 5.5.5.5.19: In Exercises 1922, use a doubleangle formula to rewrite the expres...
 5.5.5.5.20: In Exercises 1922, use a doubleangle formula to rewrite the expres...
 5.5.5.5.21: In Exercises 1922, use a doubleangle formula to rewrite the expres...
 5.5.5.5.22: In Exercises 1922, use a doubleangle formula to rewrite the expres...
 5.5.5.5.23: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.24: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.25: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.26: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.27: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.28: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.29: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.30: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.31: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.32: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.33: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.34: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.35: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.36: In Exercises 2336, rewrite the expression in terms of the first pow...
 5.5.5.5.37: In Exercises 37 and 38, use the figure to find the exact value of e...
 5.5.5.5.38: In Exercises 37 and 38, use the figure to find the exact value of e...
 5.5.5.5.39: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.40: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.41: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.42: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.43: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.44: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.45: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.46: In Exercises 3946, use the halfangle formulas to determine the exa...
 5.5.5.5.47: In Exercises 4752, find the exact values of and using the halfangl...
 5.5.5.5.48: In Exercises 4752, find the exact values of and using the halfangl...
 5.5.5.5.49: In Exercises 4752, find the exact values of and using the halfangl...
 5.5.5.5.50: In Exercises 4752, find the exact values of and using the halfangl...
 5.5.5.5.51: In Exercises 4752, find the exact values of and using the halfangl...
 5.5.5.5.52: In Exercises 4752, find the exact values of and using the halfangl...
 5.5.5.5.53: In Exercises 5356, use the halfangle formulas to simplify the expr...
 5.5.5.5.54: In Exercises 5356, use the halfangle formulas to simplify the expr...
 5.5.5.5.55: In Exercises 5356, use the halfangle formulas to simplify the expr...
 5.5.5.5.56: In Exercises 5356, use the halfangle formulas to simplify the expr...
 5.5.5.5.57: In Exercises 5760, find the solutions of the equation in the interv...
 5.5.5.5.58: In Exercises 5760, find the solutions of the equation in the interv...
 5.5.5.5.59: In Exercises 5760, find the solutions of the equation in the interv...
 5.5.5.5.60: In Exercises 5760, find the solutions of the equation in the interv...
 5.5.5.5.61: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.62: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.63: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.64: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.65: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.66: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.67: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.68: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.69: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.70: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.71: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.72: In Exercises 6172, use the producttosum formulas to write the pro...
 5.5.5.5.73: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.74: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.75: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.76: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.77: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.78: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.79: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.80: In Exercises 7380, use the sumtoproduct formulas to write the sum...
 5.5.5.5.81: In Exercises 8184, use the sumtoproduct formulas to find the exac...
 5.5.5.5.82: In Exercises 8184, use the sumtoproduct formulas to find the exac...
 5.5.5.5.83: In Exercises 8184, use the sumtoproduct formulas to find the exac...
 5.5.5.5.84: In Exercises 8184, use the sumtoproduct formulas to find the exac...
 5.5.5.5.85: In Exercises 8588, find the solutions of the equation in the interv...
 5.5.5.5.86: In Exercises 8588, find the solutions of the equation in the interv...
 5.5.5.5.87: In Exercises 8588, find the solutions of the equation in the interv...
 5.5.5.5.88: In Exercises 8588, find the solutions of the equation in the interv...
 5.5.5.5.89: In Exercises 8992, use the figure and trigonometric identities to f...
 5.5.5.5.90: In Exercises 8992, use the figure and trigonometric identities to f...
 5.5.5.5.91: In Exercises 8992, use the figure and trigonometric identities to f...
 5.5.5.5.92: In Exercises 8992, use the figure and trigonometric identities to f...
 5.5.5.5.93: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.94: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.95: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.96: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.97: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.98: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.99: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.100: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.101: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.102: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.103: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.104: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.105: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.106: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.107: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.108: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.109: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.110: In Exercises 93110, verify the identity algebraically. Use a graphi...
 5.5.5.5.111: In Exercises 111114, rewrite the function using the powerreducing f...
 5.5.5.5.112: In Exercises 111114, rewrite the function using the powerreducing f...
 5.5.5.5.113: In Exercises 111114, rewrite the function using the powerreducing f...
 5.5.5.5.114: In Exercises 111114, rewrite the function using the powerreducing f...
 5.5.5.5.115: In Exercises 115120, write the trigonometric expression as an algeb...
 5.5.5.5.116: In Exercises 115120, write the trigonometric expression as an algeb...
 5.5.5.5.117: In Exercises 115120, write the trigonometric expression as an algeb...
 5.5.5.5.118: In Exercises 115120, write the trigonometric expression as an algeb...
 5.5.5.5.119: In Exercises 115120, write the trigonometric expression as an algeb...
 5.5.5.5.120: In Exercises 115120, write the trigonometric expression as an algeb...
 5.5.5.5.121: In Exercises 121124, (a) use a graphing utility to graph the functi...
 5.5.5.5.122: In Exercises 121124, (a) use a graphing utility to graph the functi...
 5.5.5.5.123: In Exercises 121124, (a) use a graphing utility to graph the functi...
 5.5.5.5.124: In Exercises 121124, (a) use a graphing utility to graph the functi...
 5.5.5.5.125: In Exercises 125 and 126, the graph of a function f is shown over t...
 5.5.5.5.126: In Exercises 125 and 126, the graph of a function f is shown over t...
 5.5.5.5.127: Projectile Motion The range of a projectile fired at an angle with ...
 5.5.5.5.128: Geometry The length of each of the two equal sides of an isosceles ...
 5.5.5.5.129: Railroad Track When two railroad tracks merge, the overlapping port...
 5.5.5.5.130: Mach Number The mach number of an airplane is the ratio of its spee...
 5.5.5.5.131: True or False? In Exercises 131 and 132, determine whether the stat...
 5.5.5.5.132: True or False? In Exercises 131 and 132, determine whether the stat...
 5.5.5.5.133: Conjecture Consider the function (a) Use a graphing utility to grap...
 5.5.5.5.134: Exploration Consider the function (a) Use the powerreducing formul...
 5.5.5.5.135: Writing Describe how you can use a doubleangle formula or a halfa...
 5.5.5.5.136: (a) Write a formula for cos 3. (b) Write a formula for cos 4.
 5.5.5.5.137: In Exercises 137140, (a) plot the points, (b) find the distance bet...
 5.5.5.5.138: In Exercises 137140, (a) plot the points, (b) find the distance bet...
 5.5.5.5.139: In Exercises 137140, (a) plot the points, (b) find the distance bet...
 5.5.5.5.140: In Exercises 137140, (a) plot the points, (b) find the distance bet...
 5.5.5.5.141: In Exercises 141144, find (if possible) the complement and suppleme...
 5.5.5.5.142: In Exercises 141144, find (if possible) the complement and suppleme...
 5.5.5.5.143: In Exercises 141144, find (if possible) the complement and suppleme...
 5.5.5.5.144: In Exercises 141144, find (if possible) the complement and suppleme...
 5.5.5.5.145: Find the radian measure of the central angle of a circle with a rad...
 5.5.5.5.146: Find the length of the arc on a circle of radius 21 centimeters int...
 5.5.5.5.147: In Exercises 147150, sketch a graph of the function. (Include two f...
 5.5.5.5.148: In Exercises 147150, sketch a graph of the function. (Include two f...
 5.5.5.5.149: In Exercises 147150, sketch a graph of the function. (Include two f...
 5.5.5.5.150: In Exercises 147150, sketch a graph of the function. (Include two f...
Solutions for Chapter 5.5: MultipleAngle and ProducttoSum Formulas
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 5.5: MultipleAngle and ProducttoSum Formulas
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 150 problems in chapter 5.5: MultipleAngle and ProducttoSum Formulas have been answered, more than 46014 students have viewed full stepbystep solutions from this chapter. Chapter 5.5: MultipleAngle and ProducttoSum Formulas includes 150 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.