 7.3.1: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.2: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.3: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.4: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.5: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.6: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.7: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.8: In Exercises 18, fill in the blank(s). 1. A _______ can be used to ...
 7.3.9: What two characteristics determine whether two directed line segmen...
 7.3.10: What do you call a vector that has a magnitude of 1?
 7.3.11: In Exercises 11 and 12, show that u and v are equivalent.
 7.3.12: In Exercises 11 and 12, show that u and v are equivalent.
 7.3.13: In Exercises 1324, find the component form and the magnitude of the...
 7.3.14: In Exercises 1324, find the component form and the magnitude of the...
 7.3.15: In Exercises 1324, find the component form and the magnitude of the...
 7.3.16: In Exercises 1324, find the component form and the magnitude of the...
 7.3.17: In Exercises 1324, find the component form and the magnitude of the...
 7.3.18: In Exercises 1324, find the component form and the magnitude of the...
 7.3.19: In Exercises 1324, find the component form and the magnitude of the...
 7.3.20: In Exercises 1324, find the component form and the magnitude of the...
 7.3.21: In Exercises 1324, find the component form and the magnitude of the...
 7.3.22: In Exercises 1324, find the component form and the magnitude of the...
 7.3.23: In Exercises 1324, find the component form and the magnitude of the...
 7.3.24: In Exercises 1324, find the component form and the magnitude of the...
 7.3.25: In Exercises 2530, use the figure to sketch a graph of the specifie...
 7.3.26: In Exercises 2530, use the figure to sketch a graph of the specifie...
 7.3.27: In Exercises 2530, use the figure to sketch a graph of the specifie...
 7.3.28: In Exercises 2530, use the figure to sketch a graph of the specifie...
 7.3.29: In Exercises 2530, use the figure to sketch a graph of the specifie...
 7.3.30: In Exercises 2530, use the figure to sketch a graph of the specifie...
 7.3.31: In Exercises 3136, use the figure to sketch a graph of the specifie...
 7.3.32: In Exercises 3136, use the figure to sketch a graph of the specifie...
 7.3.33: In Exercises 3136, use the figure to sketch a graph of the specifie...
 7.3.34: In Exercises 3136, use the figure to sketch a graph of the specifie...
 7.3.35: In Exercises 3136, use the figure to sketch a graph of the specifie...
 7.3.36: In Exercises 3136, use the figure to sketch a graph of the specifie...
 7.3.37: In Exercises 37 42, find (a) u + v, (b) u v, (c) 2u 3v, and (d) 1 2...
 7.3.38: In Exercises 37 42, find (a) u + v, (b) u v, (c) 2u 3v, and (d) 1 2...
 7.3.39: In Exercises 37 42, find (a) u + v, (b) u v, (c) 2u 3v, and (d) 1 2...
 7.3.40: In Exercises 37 42, find (a) u + v, (b) u v, (c) 2u 3v, and (d) 1 2...
 7.3.41: In Exercises 37 42, find (a) u + v, (b) u v, (c) 2u 3v, and (d) 1 2...
 7.3.42: In Exercises 37 42, find (a) u + v, (b) u v, (c) 2u 3v, and (d) 1 2...
 7.3.43: In Exercises 4346, use the figure to write the vector in terms of t...
 7.3.44: In Exercises 4346, use the figure to write the vector in terms of t...
 7.3.45: In Exercises 4346, use the figure to write the vector in terms of t...
 7.3.46: In Exercises 4346, use the figure to write the vector in terms of t...
 7.3.47: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.48: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.49: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.50: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.51: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.52: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.53: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.54: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.55: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.56: In Exercises 4756, find a unit vector in the direction of the given...
 7.3.57: In Exercises 5762, find the vector v with the given magnitude and t...
 7.3.58: In Exercises 5762, find the vector v with the given magnitude and t...
 7.3.59: In Exercises 5762, find the vector v with the given magnitude and t...
 7.3.60: In Exercises 5762, find the vector v with the given magnitude and t...
 7.3.61: In Exercises 5762, find the vector v with the given magnitude and t...
 7.3.62: In Exercises 5762, find the vector v with the given magnitude and t...
 7.3.63: In Exercises 6366, the initial and terminal points of a vector are ...
 7.3.64: In Exercises 6366, the initial and terminal points of a vector are ...
 7.3.65: In Exercises 6366, the initial and terminal points of a vector are ...
 7.3.66: In Exercises 6366, the initial and terminal points of a vector are ...
 7.3.67: In Exercises 6772, find the component form of v and sketch the spec...
 7.3.68: In Exercises 6772, find the component form of v and sketch the spec...
 7.3.69: In Exercises 6772, find the component form of v and sketch the spec...
 7.3.70: In Exercises 6772, find the component form of v and sketch the spec...
 7.3.71: In Exercises 6772, find the component form of v and sketch the spec...
 7.3.72: In Exercises 6772, find the component form of v and sketch the spec...
 7.3.73: n Exercises 7378, find the magnitude and direction angle of the vec...
 7.3.74: n Exercises 7378, find the magnitude and direction angle of the vec...
 7.3.75: n Exercises 7378, find the magnitude and direction angle of the vec...
 7.3.76: n Exercises 7378, find the magnitude and direction angle of the vec...
 7.3.77: n Exercises 7378, find the magnitude and direction angle of the vec...
 7.3.78: n Exercises 7378, find the magnitude and direction angle of the vec...
 7.3.79: In Exercises 7986, find the component form of v given its magnitude...
 7.3.80: In Exercises 7986, find the component form of v given its magnitude...
 7.3.81: In Exercises 7986, find the component form of v given its magnitude...
 7.3.82: In Exercises 7986, find the component form of v given its magnitude...
 7.3.83: In Exercises 7986, find the component form of v given its magnitude...
 7.3.84: In Exercises 7986, find the component form of v given its magnitude...
 7.3.85: In Exercises 7986, find the component form of v given its magnitude...
 7.3.86: In Exercises 7986, find the component form of v given its magnitude...
 7.3.87: In Exercises 8790, find the component form of the sum of u and v wi...
 7.3.88: In Exercises 8790, find the component form of the sum of u and v wi...
 7.3.89: In Exercises 8790, find the component form of the sum of u and v wi...
 7.3.90: In Exercises 8790, find the component form of the sum of u and v wi...
 7.3.91: In Exercises 91 and 92, use the Law of Cosines to find the angle be...
 7.3.92: In Exercises 91 and 92, use the Law of Cosines to find the angle be...
 7.3.93: In Exercises 93 and 94, graph the vectors and the resultant of the ...
 7.3.94: In Exercises 93 and 94, graph the vectors and the resultant of the ...
 7.3.95: In Exercises 95 and 96, find the angle between the forces given the...
 7.3.96: In Exercises 95 and 96, find the angle between the forces given the...
 7.3.97: A ball is thrown with an initial velocity of 70 feet per second, at...
 7.3.98: A gun with a muzzle velocity of 1200 feet per second is fired at an...
 7.3.99: The cranes shown in the figure are lifting an object that weighs 20...
 7.3.100: Use the figure to determine the tension in each cable supporting th...
 7.3.101: A loaded barge is being towed by two tugboats, and the magnitude of...
 7.3.102: To carry a 100pound cylindrical weight, two people lift on the end...
 7.3.103: Forces with magnitudes of 150 newtons and 220 newtons act on a hook...
 7.3.104: A tetherball weighing 1 pound is pulled outward from the pole by a ...
 7.3.105: An airplane is flying in the direction 148 with an airspeed of 860 ...
 7.3.106: A commercial jet is flying from Miami to Seattle. The jets velocity...
 7.3.107: In Exercises 107110, determine whether the statement is true or fal...
 7.3.108: In Exercises 107110, determine whether the statement is true or fal...
 7.3.109: In Exercises 107110, determine whether the statement is true or fal...
 7.3.110: In Exercises 107110, determine whether the statement is true or fal...
 7.3.111: Consider two forces of equal magnitude acting on a point. (a) If th...
 7.3.112: Consider two forces F1 = 10, 0 and F2 = 5cos , sin . (a) Write F1 +...
 7.3.113: Give geometric descriptions of (a) vector addition and (b) scalar m...
 7.3.114: Use the figure to determine whether each statement is true or false...
 7.3.115: Prove that (cos )i + (sin )j is a unit vector for any value of .
 7.3.116: Write a program for a graphing utility that graphs two vectors and ...
 7.3.117: In Exercises 117 and 118, use the program in Exercise 116 to find t...
 7.3.118: In Exercises 117 and 118, use the program in Exercise 116 to find t...
 7.3.119: Identify each quantity as either a scalar or a vector. Explain. (a)...
 7.3.120: In Exercises 120 125, simplify the expression. 120. ( 6x4 7y2)(14x1...
 7.3.121: In Exercises 120 125, simplify the expression. 120. ( 6x4 7y2)(14x1...
 7.3.122: In Exercises 120 125, simplify the expression. 120. ( 6x4 7y2)(14x1...
 7.3.123: In Exercises 120 125, simplify the expression. 120. ( 6x4 7y2)(14x1...
 7.3.124: In Exercises 120 125, simplify the expression. 120. ( 6x4 7y2)(14x1...
 7.3.125: In Exercises 120 125, simplify the expression. 120. ( 6x4 7y2)(14x1...
 7.3.126: In Exercises 126129, solve the equation. 126. cos x(cos x + 1) = 0 ...
 7.3.127: In Exercises 126129, solve the equation. 126. cos x(cos x + 1) = 0 ...
 7.3.128: In Exercises 126129, solve the equation. 126. cos x(cos x + 1) = 0 ...
 7.3.129: In Exercises 126129, solve the equation. 126. cos x(cos x + 1) = 0 ...
 7.3.130: In Exercises 130 and 131, use the trigonometric substitution to wri...
 7.3.131: In Exercises 130 and 131, use the trigonometric substitution to wri...
Solutions for Chapter 7.3: Additional Topics in Trigonometry
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 7.3: Additional Topics in Trigonometry
Get Full SolutionsSince 131 problems in chapter 7.3: Additional Topics in Trigonometry have been answered, more than 58859 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. Chapter 7.3: Additional Topics in Trigonometry includes 131 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

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

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.

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

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·