 4.1: Concept Check Which one of the following is true about the graph of...
 4.2: Concept Check Which one of the following is false about the graph o...
 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: Concept Check Identify the circular function that satisfies each de...
 4.18: Concept Check Identify the circular function that satisfies each de...
 4.19: Concept Check Identify the circular function that satisfies each de...
 4.20: Concept Check Identify the circular function that satisfies each de...
 4.21: Concept Check Identify the circular function that satisfies each de...
 4.22: Concept Check Identify the circular function that satisfies each de...
 4.23: Concept Check Identify the circular function that satisfies each de...
 4.24: Concept Check Identify the circular function that satisfies each de...
 4.25: Graph each function over a oneperiod interval. y = 3 sin x
 4.26: Graph each function over a oneperiod interval. y = 1 2 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 = 1 2 cot 3x
 4.35: Graph each function over a oneperiod interval. y = cos ax  p 4 b
 4.36: Graph each function over a oneperiod interval. y = tan ax  p 2 b
 4.37: Graph each function over a oneperiod interval. y = sec a2x + p 3 b
 4.38: Graph each function over a oneperiod interval. y = sin a3x + p 2 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 =  1 2 cos1px  p2
 4.43: Concept Check Determine the range of a function of the form 1x2 = 2...
 4.44: Concept Check Determine the range of a function of the form 1x2 = 2...
 4.45: Connecting Graphs with Equations Determine the simplest form of an ...
 4.46: Connecting Graphs with Equations Determine the simplest form of an ...
 4.47: Connecting Graphs with Equations Determine the simplest form of an ...
 4.48: Connecting Graphs with Equations Determine the simplest form of an ...
 4.49: Viewing Angle to an Object Let a person whose eyes are h1 feet from...
 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 s(t) inch...
 4.56: An object in simple harmonic motion has position function s(t) 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: Review Exercises
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 4: Review Exercises
Get Full SolutionsChapter 4: Review Exercises includes 58 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9780321671776. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Since 58 problems in chapter 4: Review Exercises have been answered, more than 35930 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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

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.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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