- 6.1.1: ________ means measurement of triangles.
- 6.1.2: An ________ is determined by rotating a ray about its endpoint.
- 6.1.3: Two angles that have the same initial and terminal sides are ________.
- 6.1.4: The angle measure that is equivalent to of a complete revolution ab...
- 6.1.5: Angles with measures between and are ________ angles, and angles wi...
- 6.1.6: Two positive angles that have a sum of are ________ angles, whereas...
- 6.1.7: One ________ is the measure of a central angle that intercepts an a...
- 6.1.8: The ________ speed of a particle is the ratio of the arc length tra...
- 6.1.9: The ________ speed of a particle is the ratio of the change in the ...
- 6.1.10: The area of a sector of a circle with radius and central angle wher...
- 6.1.15: (a) (b)
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- 6.1.34: (a) 0.355 (b)
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- 6.1.49: (a) (b)
- 6.1.50: (a) 6.02 (b) 2.25
- 6.1.51: (a) (b)
- 6.1.52: (a) (b)
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- 6.1.54: (a) 4 (b)
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- 6.1.69: (a) (b)
- 6.1.70: (a) (b)
- 6.1.71: 45
- 6.1.72: 87.4
- 6.1.73: 216.35
- 6.1.74: 48.27
- 6.1.75: 532
- 6.1.76: 345
- 6.1.77: 0.83
- 6.1.78: 0.54
- 6.1.79: 7
- 6.1.80: 5 11
- 6.1.81: 15 8
- 6.1.82: 13 2
- 6.1.83: 2
- 6.1.84: 0.57 1
- 6.1.85: 5
- 6.1.86: 10 29
- 6.1.87: 32
- 6.1.88: 75 60
- 6.1.89: 4 inches 18 inches
- 6.1.90: 14 feet 8 feet
- 6.1.91: 14.5 centimeters 25 centimeters
- 6.1.92: 80 kilometers 160 kilometers
- 6.1.93: 15 inches
- 6.1.94: 9 feet
- 6.1.95: 3 meters
- 6.1.96: 20 centimeters
- 6.1.97: 6 inches
- 6.1.98: 12 millimeters
- 6.1.99: 2.5 feet
- 6.1.100: 1.4 miles
- 6.1.101: Dallas, Texas Omaha, Nebraska
- 6.1.102: San Francisco, California Seattle, Washington
- 6.1.103: DIFFERENCE IN LATITUDES Assuming that Earth is a sphere of radius 6...
- 6.1.104: DIFFERENCE IN LATITUDES Assuming that Earth is a sphere of radius 6...
- 6.1.105: INSTRUMENTATION The pointer on a voltmeter is 6 centimeters in leng...
- 6.1.106: ELECTRIC HOIST An electric hoist is being used to lift a beam (see ...
- 6.1.107: ANGULAR SPEED A car is moving at a rate of 65 miles per hour, and t...
- 6.1.108: ANGULAR SPEED A two-inch-diameter pulley on an electric motor that ...
- 6.1.109: LINEAR AND ANGULAR SPEED A -inch circular power saw blade rotates a...
- 6.1.110: LINEAR AND ANGULAR SPEED A carousel with a 50-foot diameter makes 4...
- 6.1.111: LINEAR AND ANGULAR SPEED The diameter of a DVD is approximately 12 ...
- 6.1.112: ANGULAR SPEED A computerized spin balance machine rotates a 25-inch...
- 6.1.113: AREA A sprinkler on a golf green is set to spray water over a dista...
- 6.1.114: AREA A cars rear windshield wiper rotates The total length of the w...
- 6.1.115: AREA A sprinkler system on a farm is set to spray water over a dist...
- 6.1.116: SPEED OF A BICYCLE The radii of the pedal sprocket, the wheel sproc...
- 6.1.117: A measurement of 4 radians corresponds to two complete revolutions ...
- 6.1.118: The difference between the measures of two coterminal angles is alw...
- 6.1.119: An angle that measures lies in Quadrant III.
- 6.1.120: CAPSTONE Write a short paragraph in your own words explaining the m...
- 6.1.121: THINK ABOUT IT A fan motor turns at a given angular speed. How does...
- 6.1.122: THINK ABOUT IT Is a degree or a radian the larger unit of measure? ...
- 6.1.123: WRITING If the radius of a circle is increasing and the magnitude o...
- 6.1.124: PROOF Prove that the area of a circular sector of radius with centr...
Solutions for Chapter 6.1: Angles and Their Measure
Full solutions for Algebra and Trigonometry | 8th Edition
peA) = det(A - AI) has peA) = zero matrix.
Remove row i and column j; multiply the determinant by (-I)i + j •
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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!
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.