 5.5.1: In Exercises 1 and 2, fill in the blank. 1. The graphs of the tange...
 5.5.2: In Exercises 1 and 2, fill in the blank. 2. To sketch the graph of ...
 5.5.3: Which two parent trigonometric functions have a period of and a ran...
 5.5.4: What is the damping factor of the function f(x) = e2x sin x?
 5.5.5: In Exercises 58, use the graph of the function to answer each quest...
 5.5.6: In Exercises 58, use the graph of the function to answer each quest...
 5.5.7: In Exercises 58, use the graph of the function to answer each quest...
 5.5.8: In Exercises 58, use the graph of the function to answer each quest...
 5.5.9: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.10: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.11: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.12: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.13: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.14: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.15: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.16: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.17: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.18: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.19: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.20: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.21: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.22: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.23: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.24: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.25: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.26: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.27: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.28: In Exercises 928, sketch the graph of the function. (Include two fu...
 5.5.29: In Exercises 2934, use a graphing utility to graph the function (in...
 5.5.30: In Exercises 2934, use a graphing utility to graph the function (in...
 5.5.31: In Exercises 2934, use a graphing utility to graph the function (in...
 5.5.32: In Exercises 2934, use a graphing utility to graph the function (in...
 5.5.33: In Exercises 2934, use a graphing utility to graph the function (in...
 5.5.34: In Exercises 2934, use a graphing utility to graph the function (in...
 5.5.35: In Exercises 35 40, use a graph of the function to approximate the ...
 5.5.36: In Exercises 35 40, use a graph of the function to approximate the ...
 5.5.37: In Exercises 35 40, use a graph of the function to approximate the ...
 5.5.38: In Exercises 35 40, use a graph of the function to approximate the ...
 5.5.39: In Exercises 35 40, use a graph of the function to approximate the ...
 5.5.40: In Exercises 35 40, use a graph of the function to approximate the ...
 5.5.41: In Exercises 41 46, use the graph of the function to determine whet...
 5.5.42: In Exercises 41 46, use the graph of the function to determine whet...
 5.5.43: In Exercises 41 46, use the graph of the function to determine whet...
 5.5.44: In Exercises 41 46, use the graph of the function to determine whet...
 5.5.45: In Exercises 41 46, use the graph of the function to determine whet...
 5.5.46: In Exercises 41 46, use the graph of the function to determine whet...
 5.5.47: In Exercises 47 50, use a graphing utility to graph the two equatio...
 5.5.48: In Exercises 47 50, use a graphing utility to graph the two equatio...
 5.5.49: In Exercises 47 50, use a graphing utility to graph the two equatio...
 5.5.50: In Exercises 47 50, use a graphing utility to graph the two equatio...
 5.5.51: In Exercises 5154, match the function with its graph. Describe the ...
 5.5.52: In Exercises 5154, match the function with its graph. Describe the ...
 5.5.53: In Exercises 5154, match the function with its graph. Describe the ...
 5.5.54: In Exercises 5154, match the function with its graph. Describe the ...
 5.5.55: In Exercises 55 58, use a graphing utility to graph the function an...
 5.5.56: In Exercises 55 58, use a graphing utility to graph the function an...
 5.5.57: In Exercises 55 58, use a graphing utility to graph the function an...
 5.5.58: In Exercises 55 58, use a graphing utility to graph the function an...
 5.5.59: In Exercises 59 and 60, use a graphing utility to graph the functio...
 5.5.60: In Exercises 59 and 60, use a graphing utility to graph the functio...
 5.5.61: In Exercises 61 and 62, use a graphing utility to graph the functio...
 5.5.62: In Exercises 61 and 62, use a graphing utility to graph the functio...
 5.5.63: A plane flying at an altitude of 7 miles over level ground will pas...
 5.5.64: A television camera is on a reviewing platform 27 meters from the s...
 5.5.65: An object weighing W pounds is suspended from a ceiling by a steel ...
 5.5.66: A crossed belt connects a 10centimeter pulley on an electric motor...
 5.5.67: The motion of an oscillating weight suspended by a spring was measu...
 5.5.68: In Exercises 6871, determine whether the statement is true or false...
 5.5.69: In Exercises 6871, determine whether the statement is true or false...
 5.5.70: In Exercises 6871, determine whether the statement is true or false...
 5.5.71: In Exercises 6871, determine whether the statement is true or false...
 5.5.72: Consider the functions f(x) = 2 sin x and g(x) = 1 2 csc x on the i...
 5.5.73: Consider the functions given by f(x) = tan x 2 and g(x) = 1 2 sec x...
 5.5.74: (a) Use a graphing utility to graph each function. y1 = 4 (sin x + ...
 5.5.75: In Exercises 75 and 76, use a graphing utility to explore the ratio...
 5.5.76: In Exercises 75 and 76, use a graphing utility to explore the ratio...
 5.5.77: Using calculus, it can be shown that the tangent function can be ap...
 5.5.78: Determine which function is represented by each graph. Do not use a...
 5.5.79: In Exercises 79 82, identify the rule of algebra illustrated by the...
 5.5.80: In Exercises 79 82, identify the rule of algebra illustrated by the...
 5.5.81: In Exercises 79 82, identify the rule of algebra illustrated by the...
 5.5.82: In Exercises 79 82, identify the rule of algebra illustrated by the...
 5.5.83: In Exercises 83 86, determine whether the function is onetoone. I...
 5.5.84: In Exercises 83 86, determine whether the function is onetoone. I...
 5.5.85: In Exercises 83 86, determine whether the function is onetoone. I...
 5.5.86: In Exercises 83 86, determine whether the function is onetoone. I...
 5.5.87: In Exercises 8790, identify the domain, any intercepts, and any asy...
 5.5.88: In Exercises 8790, identify the domain, any intercepts, and any asy...
 5.5.89: In Exercises 8790, identify the domain, any intercepts, and any asy...
 5.5.90: In Exercises 8790, identify the domain, any intercepts, and any asy...
Solutions for Chapter 5.5: Trigonometric Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 5.5: Trigonometric Functions
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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).

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.