 4.1.Problem 1: Graph y sin x.
 4.1.a: How do we use the unit circle to graph the function y sin x?
 4.1.1: The graph of the sine function illustrates how the ____________ of ...
 4.1.Problem 2: Graph y sec x.
 4.1.b: What is the period of a function?
 4.1.2: For a periodic function f, the period p is the ___________ positive...
 4.1.Problem 3: Show that secant is an even function
 4.1.c: What is the definition of an even function?
 4.1.3: To calculate the amplitude for a function, take half the __________...
 4.1.Problem 4: Find exact values for each of the following. a. sin 5 6 b. sec (150) G
 4.1.d: What type of symmetry will the graph of an odd function have?
 4.1.4: If c is a domain value for a function f and f(c) 0, then x c is cal...
 4.1.5: A function is even if the opposite input results in an ________ out...
 4.1.6: The graph of an even function is symmetric about the _______ and th...
 4.1.7: Which trigonometric functions have a defined amplitude?
 4.1.8: Which trigonometric functions have asymptotes?
 4.1.9: Which trigonometric functions do not have real zeros?
 4.1.10: Which trigonometric
 4.1.11: Which trigonometric
 4.1.12: Which trigonometric
 4.1.13: y cos x
 4.1.14: y cot x
 4.1.15: y csc x
 4.1.16: y sin x
 4.1.17: y tan x
 4.1.18: y sec x
 4.1.19: cos x
 4.1.20: y cot x
 4.1.21: y csc x
 4.1.22: y sin x
 4.1.23: y tan x
 4.1.24: y sec x
 4.1.25: sin x 0
 4.1.26: cos x 0
 4.1.27: sin x 1 2
 4.1.28: cos x 1
 4.1.29: tan x 0
 4.1.30: cot x 0
 4.1.31: sec x 1 3
 4.1.32: csc x 1
 4.1.33: tan x is undefined
 4.1.34: cot x is undefined
 4.1.35: csc x is undefined
 4.1.36: sec x is undefined
 4.1.37: cos (60)
 4.1.38: cos (120)
 4.1.39: cos 4
 4.1.40: cos
 4.1.41: sin (30)
 4.1.42: sin (90)
 4.1.43: sin 4
 4.1.44: sin
 4.1.45: If sin 1/3, find sin (). 46.
 4.1.46: If cos 1/3, find cos (). Mak
 4.1.47: sin (180 ) sin 48.
 4.1.48: cos (180 ) cos 49.
 4.1.49: Show that tangent is an odd function
 4.1.50: Show that cotangent is an odd function.
 4.1.51: sin () cot () cos 52. co
 4.1.52: cos () tan sin 53.
 4.1.53: sin () sec () cot () 1 54. co
 4.1.54: cos () csc () tan () 1 55. cs
 4.1.55: csc sin () 56
 4.1.56: sec cos () sin2 cos cos2
 4.1.57: cos tan sin 58.
 4.1.58: sin tan cos sec 59.
 4.1.59: (1 sin )(1 sin ) cos2 60. (
 4.1.60: (sin cos ) 2 1 2 sin cos Wri
 4.1.61:
 4.1.62:
 4.1.63:
 4.1.64:
 4.1.65: y A sin x for A 1, 2, 3 6
 4.1.66: y A sin x for A 1, 1 2 , 1 3
 4.1.67: y A cos x for A 1, 0.6, 0.2 6
 4.1.68: y A cos x for A 1, 3, 5
 4.1.69: 3 sin x, y 3 sin x 70
 4.1.70: y 4 cos x, y 4 cos x U
 4.1.71: y sin Bx for B 1, 2 7
 4.1.72: y sin Bx for B 1, 4
 4.1.73: y cos Bx for B 1, 1 2 74
 4.1.74: y cos Bx for B 1, 1 3
 4.1.75: Sketch the graph of y
 4.1.76: Sketch the graph of y sec x. For which values of x, 0 x 2, is sec x...
 4.1.77: If tan 3, find tan (). a. 3 b. 3 c. 1 3 d. 1 3 78
 4.1.78: To prove cos () csc () tan () 1, which of the following is correct ...
Solutions for Chapter 4.1: Basic Graphs
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 4.1: Basic Graphs
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9781111826857. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. Since 86 problems in chapter 4.1: Basic Graphs have been answered, more than 41211 students have viewed full stepbystep solutions from this chapter. Chapter 4.1: Basic Graphs includes 86 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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!

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.