 4.1: Concept Check Which one of the following statements is true about t...
 4.2: Concept Check Which one of the following statements is false about ...
 4.3: Concept Check Which of the basic circular functions can have yvalu...
 4.4: Concept Check Which of the basic circular functions can have yvalu...
 4.5: For each function, give the amplitude, period, vertical translation...
 4.6: For each function, give the amplitude, period, vertical translation...
 4.7: For each function, give the amplitude, period, vertical translation...
 4.8: For each function, give the amplitude, period, vertical translation...
 4.9: For each function, give the amplitude, period, vertical translation...
 4.10: For each function, give the amplitude, period, vertical translation...
 4.11: For each function, give the amplitude, period, vertical translation...
 4.12: For each function, give the amplitude, period, vertical translation...
 4.13: For each function, give the amplitude, period, vertical translation...
 4.14: For each function, give the amplitude, period, vertical translation...
 4.15: For each function, give the amplitude, period, vertical translation...
 4.16: For each function, give the amplitude, period, vertical translation...
 4.17: Identify the circular function that satisfies each description peri...
 4.18: Identify the circular function that satisfies each description peri...
 4.19: Identify the circular function that satisfies each description peri...
 4.20: Identify the circular function that satisfies each description peri...
 4.21: Identify the circular function that satisfies each description peri...
 4.22: Identify the circular function that satisfies each description peri...
 4.23: Provide a short explanation Suppose that defines a sine function wi...
 4.24: Provide a short explanation Suppose that defines a sine function wi...
 4.25: Graph each function over a oneperiod interval. y = 3 sin x
 4.26: Graph each function over a oneperiod interval. y = 12 sec x
 4.27: Graph each function over a oneperiod interval. . y = tan x
 4.28: Graph each function over a oneperiod interval. y = 2 cos x
 4.29: Graph each function over a oneperiod interval. y = 2 + cot x
 4.30: Graph each function over a oneperiod interval. y = 1 + csc x
 4.31: Graph each function over a oneperiod interval. y = sin 2x
 4.32: Graph each function over a oneperiod interval. . y = tan 3x
 4.33: Graph each function over a oneperiod interval. y = 3 cos 2x
 4.34: Graph each function over a oneperiod interval. y = 12 cot 3x
 4.35: Graph each function over a oneperiod interval. y = cos ax  p4 b
 4.36: Graph each function over a oneperiod interval. y = tan ax  p2 b
 4.37: Graph each function over a oneperiod interval. y = sec a2x + p3 b
 4.38: Graph each function over a oneperiod interval. y = sin a3x + p2 b
 4.39: Graph each function over a oneperiod interval. y = 1 + 2 cos 3x
 4.40: Graph each function over a oneperiod interval. y = 1  3 sin 2x
 4.41: Graph each function over a oneperiod interval. y = 2 sin px
 4.42: Graph each function over a oneperiod interval. y =  12 cos1px  p2
 4.43: A set of temperature data (in F) is given for a particular location...
 4.44: A set of temperature data (in F) is given for a particular location...
 4.45: Determine the simplest form of an equation for each graph. Choose b...
 4.46: Determine the simplest form of an equation for each graph. Choose b...
 4.47: Determine the simplest form of an equation for each graph. Choose b...
 4.48: Determine the simplest form of an equation for each graph. Choose b...
 4.49: Viewing Angle to an Object Suppose that a person whose eyes are h1 ...
 4.50: (Modeling) Tides The figure shows a function that models the tides ...
 4.51: (Modeling) Maximum Temperatures The maximum afternoon temperature (...
 4.52: (Modeling) Average Monthly Temperature The average monthly temperat...
 4.53: (Modeling) Pollution Trends The amount of pollution in the air is l...
 4.54: (Modeling) Lynx and Hare Populations The figure shows the populatio...
 4.55: An object in simple harmonic motion has position function s1t2 inch...
 4.56: An object in simple harmonic motion has position function s1t2 inch...
 4.57: In Exercise 55, what does the frequency represent? Find the positio...
 4.58: In Exercise 56, what does the period represent? What does the ampli...
Solutions for Chapter 4: Graphs of the Circular Functions
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 4: Graphs of the Circular Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Since 58 problems in chapter 4: Graphs of the Circular Functions have been answered, more than 24104 students have viewed full stepbystep solutions from this chapter. Chapter 4: Graphs of the Circular Functions includes 58 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.