Advanced Mathematics

737 Answers

problem set i. 19 3. 2720 , 5. -2 1 7. x = -2; y = 3 9. (a) 0 or 2 (b) 0 or 2
120 3 1 y
34
2 2

11. Upper bound = 2; lower bound = -3 13. -2, 1 15.

17. x > 1 19. 181 21. 32
90

23. sec2 [ — - 6 ) + esc2 f — - 9 = esc2 6 + sec2 0 = 1 + 1
sin2 6 cos2 6

cos2 e + sin2e = 1i = sec2 flcsc2 0

sin20cos20 sin20cos20 p

25. / +x 27. K

29. 2 cis 45°; 2cisl05°; 2cisl65°; 2 cis 225°; 2 cis 285°; 2 cis 345°

problem set i.2 3. <85)(60)(2) m -2 if x < l
121 x-1 if 1 < x < 3
9 (191X2*) 2 if x > 3
y = -1 if x < 0 y-
) •y

y

J_

10 5 ;x= 11 13
2 10 ; y = 10

10 5.

13 -2 — 15. -30
’ 2’ 2

21. 4

738 Answers

23. (sec 8 + tan 6)2 s„ec 7 8a +, 2o sec 8 tan 8a -(, ■ .tan 2 8a = 1r— +, 2 sin=8-1- sin -—8

cos2 8 cos2 8 cos2 8

1 + 2 sin 8 + sin2 8 _ (1 + sin fl)(l + sin 8) _ 1 + sin 8

1 - sin2 8 (1 + sin 0)(1 - sin 8) 1 - sin 8

27. 30°, 120°, 210°, 300° \

29. 4/; -2^3 - 2i; 2VI - 2i

problem set k+5 , _ y
122 3. -mph 5.

7. y

y = x+1 if -oo < X < -1 y = -l if —oo < x < -2
9. (a) y = -2 if 1 < X < oo (b) - y = —x — 2
Si if -2 < x < 0
1 y = 2x if 0 < x < 1
if 1 < X < oo
A Ky = 2
15. -1, 1
X 5

A

i
II

.11 (a) 0 (b) 1 13. Upper bound = 5; lower bound = -2

19. No, 15 is a common factor.

23. -1 25. 5 < x < 10 27. 1, 1000

29. cos3x = cos (2x + x) = cos2xcosx - sin2xsin* = (2cos2x - l)cosx - 2 sin x cos x sin x
= 2 cos x - cosx - 2 cos x sin2 x = 2cos3x - cosx - 2cosx(l - cos2x)
- 2 cos x - cos x — 2 cos x + 2 cos3 x = 4 cos3 x - 3 cos x

739 Answers

problem set 1. 150 units 3. 300 mi 5. B = 46 yr; J = 36 yr
7. (a) Circle (b) Ellipse (c) Hyperbola (d) Parabola (e) Circle
123

9. (a) Ellipse (b) Hyperbola (c) Circle (d) Parabola (e) Circle

.11 13.

15. (a) y — -x — 1 if —< x < —1 y -4 if < x < -3
y = -2 if -1 < x < 1 (b) y -2 if -3 < * < 0
y = x-1 if 1 < x < °° x if 0 < x < 2
y 3 if 2 < x < °°

U 924
23.
17. (a) 0 or 2 (b)0or2 19. -2,-2, 1,- 21.
m

25. 10, — 27. 3.13 in. 29. 40°, 100°, 160°, 220°, 280°, 340c
10

problem set 7. 47 4^

124 1. 208.05 min 3. 200 atm 5. A = 10°; B = 40° + nx”il* + ny*j

y 15. /

17. (a) if -3 < jc < 1 y = -l if —oo < x < — 1
if 1 < x < 3 y = X+1 if -1 < x < 0
if 3 < x < °° (b) ■ y = X if 0 < x < 1
if 1 < x < 2
y = x-1 if 2 < x < °°
[y = -1

740 Answers

problem set

125

11. (-1.1478, 1.3173) 13. (1.3553,6.1632)

19. (a) Parabola (b) Parabola (c) Hyperbola (d) Ellipse (e) Circle

21. (a) y (b) y 2

if -2 < x < 0 > = -1 if -oo- < x < 1

25. (a) y = --1 x + ! if 0 < * < 2 y = x+i if 1 < x < 2
y=1 (b) if 2 < x < 3
if 3 < x < 00
y = 3-x

if 2 < x < 00 y=1

27. Upper bound = 1; lower bound = -3 29. 1, 1 + i, 1 - i

*

Index Additive identity Angle(s) (cont.) f,
of a matrix, 604
4 of a real number, 604 of elevation and depression, 109-110

A Age problems, 178-179 exterior, 7
Abbreviations Algorithm
of polygons, 91
AAAS, 68 definition of, 624
antilog, 428 for synthetic division, 624 greater than 360°, 256
cos, 108 least squares, 322
cot, 293 Altitude(s) interior, 7
CPCTC, 70 as an apothem, 455
esc, 293 of trapezoids, 237 of polygons, 91
deg, 281,333 of a triangle, 9
gal, 181 Amplitude, 311, 312, 375 naming of, 4
HL, 69 Analysis
hr, 179 of single-variable data obtuse, 4
iff, 58
log, 183,346 average (See mean) in polygons, 90-92
In, 346 bimodal distribution, 400
mi, 179 deviation, 397 quadrantal, 199-203
min, 179 mean, 397-400, 441, 442
rad, 277 median, 399,441 radian measure for, 277-281
rev, 354 mode, 399, 441
SAS, 68 range, 397 related, 188
sec, 293 standard deviation, 397-400
sin, 108 right, 4
SSS, 68 definition of, 397
tan, 108 of a population, 441 straight, 4
Absolute value, 160-161, 199 variance, 397
of a complex number, 413 of two-variable data supplementary, 4
function, 161 linear regression, 243-244
symbol, 160 with a calculator, 321—322 vertical, 5
of trigonometric functions, 188, 258 Angle-angle-angle-side
Abstract congruency postulate, 68 Angular
coefficients, 405 -406 Angle(s)
acute, 4 measure, 83, 84
to solve systems, 422-423 alternate, 8
equations, 124 bisectors in radians, 277-281
Acute drawing of, 38
angles, 4 and side ratios, 63-64 velocity, 354-356
triangles, 6 central, 9, 10
complementary, 4-5,467 Antilogarithm, 426-428, 583
determining, 27 congruent, 4
Addition copying, 37 definition of, 426
corresponding, 8
of matrices, 603-608 coterminal, 4, 256 Apothem, 455
rule for equations, 124 definitions of, 3-4
of trigonometric functions, 256-258 Arccos, 223, 224,614
of vectors, 205-207
Arccotan, 615-616

Arccsc, 614-615

Archimedes, 23

Arc(s)

definition of, 9

length, 281

major, 83

measure of, 83, 84, 277

minor, 83

Arcsec, 614-615

Arcsin, 225, 226, 613-614

Arctan, 228, 330, 615-616

Area(s)

of circles, 9

of rectangles, 9, 14

of regular polygons, 456

of segments, 369

of similar figures, 45-46

of trapezoids, 238, 368

741

742 Index

Area(s) (cont.) c Cones, 19
of triangles, 9, 14-15, 367-368 circular, 20, 301
with sine, 554 Calculators » as pyramids, 19
right, 19-21
Arguments, 286-287, 349 for antilogarithms, 426-427 surface area of, 20, 21
of functions, 286, 287 volume of, 19
in logic, 55-58 for exponential problems, 521
in trigonometric equations, 350-351 Congruency
for logarithms, 346 of angles, 4
Arithmetic of arcs, 83
means, 513, 514, 567 Central angle, 9, 10 of circles, 83
progressions, 513, 535 definition of, 2
common difference of, 535 Chord(s), 83. See also Circle(s) of figures, 14, 67-69
notation for, 565 of line segments, 2
sequence, 513-515, 586, 587 (See products of segments, 85, 101-102 marks, 5
also progressions) postulates, 68-69
series, 586-587 Chord-tangent theorem, 93-94 symbol, 2

Arrangement. See Permutations Circle(s), 9-10, 83-87, 301-303, 445 Conic sections, 301. See also Circle(s)
Assumptions, 5-6, 115 ellipses, 445-448
Asymptotes, 162, 476 arcs of, 9, 83-84 degenerate, 671
equations of, 596, 671-673
of hyperbolas, 476, 477, 596-598 areas of, 9 hyperbolas, 475-478
of rational functions, 666 parabolas, 358-361
vertical, 162 as conic sections, 301 translations of, 595
Average. See also Mean
in rate problems, 273, 274 equations of, 409, 671 Conjugate(s)
Axiom(s), 115 of denominators, 78, 79
reflexive, 116 general form, 302, 409 pairs, 324 v

B standard form, 302, 409, 671 Constants
in summation, 242
Base(s) intersected by secants, 102-105
angles Construction, 36-40
of isosceles trapezoids, 238 intersected by tangents, 98-101 of parallels, 40
in antilogarithms, 426-427 of perpendiculars, 38-39
of cones and pyramids, 19 locus definition of, 83, 301
of cylinders and prisms, 16 parts of, 83 Coordinate(s)
of exponentials, 141, 166, 495 change in, 249, 250
m graphing, 168, 304, 419 properties of, 84-87 systems
extension of, 9, 368 Cartesian (See rectangular)
in logarithmic inequalities sectors of, 9, 369 polar, 413
base > 1, 618 rectangular, 412
base < 1, 618-619 segments of, 369
of logarithms Coplanar, 2
change of, 560-561 Circular Cosecant, 293, 468, 469
common (10), 346, 495 Cosine(s), 468, 489, 491, 517, 570-571
in equations, 183, 184 cones, 20, 301
natural (e), 346, 495 function, 308
of powers, 166 cylinders, 16 graph, 311, 312
of trapezoids, 237 law of, 489-493
of triangles, 9, 14-15 permutations, 363-364 as ratio of sides, 108, 197, 198
Cotangent, 293, 468, 469
Betweenness, 5 Circumference, 9, 83 Coterminal, 4, 256
in assumptions, 6 Cramer, 458
Clock problems, 510-511 Cramer’s rule, 458-459
Bimodal in solving systems, 575-577
distribution, 400 Closed surfaces, 16 and zero determinants, 575
Curves, 1
Binomial(s) Coefficient matrix, 458, 459, 656 closed, 16
in completing the square, 79, 80 open, 16
expansion, 472, 580-581 Coefficient(s) Cylinder(s)
squaring of, 542 circular, 16
theorem, 621 abstract, 405-406 as prisms, 16
right, 16-18
Boat-in-the-river problems, 258-259 to solve systems, 422-423 surface area of, 17-18
Box-and-whisker plots, 401-403 volume of, 17, 23
matrix, 458 Cylindrical surface, 16
first quartile, 402
interquartile range, 403 Cofactors, 590-593
outlier, 403
third quartile, 402 expansion by, 592-593

Collinear, 2, 5

Combinations

definition of, 461,462

notations for, 540

Common

difference, 535

fraction, 610

ratio, 535, 566

Commutative operation, 604

Complementary angles, 4-5, 467

Completing the square, 79-80

Complex

fractions, 122-123

numbers, 44-45, 78

absolute value of, 413

factoring over, 325-326

polar form of, 413, 551

roots of, 481-483, 551

sums and products of, 413

plane, 412-413

polynomials, 644

roots, 324-325,551

Composition of functions, 174-175

743 Index

D Elimination Exponential(s), 141, 166
'method of solving, 51-52, 139-141 equations, 166, 495-497, 521,
Decomposition of functions, 244, 245 with abstract coefficients, 458 562-564
Deductive reasoning, 54, 55, 69 of radicals, 50-51, 416 expressions, 43, 141
Degree(s) in trigonometric functions, 417 factoring of, 141-142
form, 184, 185
in a circle, 10 Ellipse(s), 445-448 functions, 166-167
mode, 108 as a conic section, 301, 445 graphing, 167-169, 304, 305, 419
in a right triangle, 4 definition of, 445, 475 growth and decay, 521-524
De Moivre, 480 equation of, 595, 671 logarithmic form of, 183-184
De Moivre’s theorem, 480 general form, 595 sketching, 167-169
Denominators standard form, 446, 475, 526, 671
rationalization of, 44, 78 foci of, 445, 527 Exponents, 43-44, 166, 426, 495
Depressed polynomials, 625, 647 major axis of, 446, 447 factoring, 141-142
Descartes’ rule of signs, 651-652 minor axis of, 446, 447 as fractions, 43, 166
Determinant(s) vertices of, 447 negative, 43
of a matrix, 438-439 simplifying of, 43-44
End points
3 X 3,575 of line segments, 1-2 Expressions
solutions, 576-577 of rays, 2 exponential, 43, 141, 142
zero, 574-575 radical, 43
Deviation Equation(s). See also Systems trigonometric, 504-505
definition of, 397 abstract, 124
Diagonals addition rule for, 124 Exterior angles, 7
in polygons, 91-92 of circles, 302, 409, 671 of a polygon, 91
of rectangles, 235 of conic sections, 595, 671-673
of rectangular solids, 46-47 of ellipses, 595, 671 F
of rhombuses, 235 exponential, 166, 495-497, 521, Faces, 16
of squares, 235 562-564 Factoring
Diameters, 83. See also Circles fractional, 49-50
Difference of hyperbolas, 475, 596-598, 671 over complex numbers, 325-326
of functions, 174 independent, 577-578 of exponents, 141-142
of two cubes, 142-143 linear, 604 trigonometric expressions, 392-394,
Direct variation. See Linear, variation forms of, 281-283
Directed distance, 249, 465 of lines, 74-77 504-505,
Directrix slope intercept, 74, 282 First quartile, 402
of a cylinder, 16 logarithmic, 184, 185, 387-390, Flex point, 629
in direction, 4 561-564 Focus (Foci)
of a parabola, 431, 432, 434 multiplication rule for, 124
Discontinuity, 162 of parabolas, 431, 434, 671 of ellipses, 445, 527
Discontinuous. See Functions, types of translated, 433 of hyperbolas, 475
Distance polynomial, 644, 647 of parabolas, 431, 434
directed, 249, 465 normalized, 638 Formula(s)
formula, 252, 262, 263 quadratic, 87-88 for area {See Area(s))
from a point to a line, 381-383 with designated roots, 272 distance, 252, 262
Division of polynomials, 124-125 radical, 50-51 midpoint, 265-266
Domain(s) roots of, 625, 644 quadratic, 87-88, 139
of a function, 152-154 trigonometric, 342-343, 392-394, for solving systems, 422-423
Dominant term, 629-630 416,508-509 for volume {See Volume(s))
Double-angle identities, 531—532, 553 Fractions
Double-intercept form, 282 Equiangular triangles, 6, 7 complex, 122-123
Equilateral triangles, 6, 7 as exponents, 43, 166
E Equilibrant, 206 as unit multipliers, 353
e (the natural base) Euclid, 115 very large, 199
Euler, 44, 458 very small, 198
finding, 427 Even functions, 465 Frequency distribution graph, 398
numerical representation of, 77, 346 Events, 499 Function(s)
Element(s) absolute value, 161
as factors, 592 independent, 501 arguments of, 286, 287
of matrices, 437, 592, 603 mutually exclusive, 538-539 combining of, 174-176
of sets, 152, 155 Experimental data of complements, 467-468
of surfaces, 16 distribution of, 398 composition of, 174-175
typical, 242 graphs of, 243, 244, 321, 322 decomposition of, 244, 245
with probability, 499 definition of, 152
discontinuous, 157
domains of, 152-154
even, 465
exponential, 166-169, 521-524

Function(s) (cont.) Geometric (cont.) Identities (cont.)
greatest integer, 663 series, 586, 588-589, 600 < trigonometric, 468-470, 517,
inverse, 289, 290 convergent, 600, 610 542-543
Arccotangent, 615 divergent, 600
Arccosine, 613 infinite, 600, 610, 611 Identity
Arccosecant, 614 function, 245
Arcsecant, 614 Geometry
Arcsine, 613 review, 1 - matrix, 607
Arctangent, 615 Image, 155
definition of, 289 Graph(s) Imaginary
logarithmic, 338, 419-420 of arccosecant, 614-615
odd, 465 of arccosine, 613-614 components, 44-45
one-to-one, 155, 157 of arcsine, 613-614 numbers, 44-45
periodic, 377 (See also Sinusoids) box-and-whisker plots, 401-403 Independent
cosecant, 546 of circles and hyperbolas, 140 equations, 577-578
cotangent, 548 that contain holes, 668-669 events, 512
secant, 546 of cosecant, 546-548 systems, 577-578
tangent, 548 of cosine, 311, 312 variables, 150
piecewise, 662-663 of cotangent, 548-549 Inductive reasoning, 54. See also Logic
poles of, 665 exponential, 167-169 Infinite sequence, 512
polynomial, 630, 632 of factored polynomials, 628-634 Inflection point, 629
products of, 174 frequency distribution, 398 Initial sides, 4
quadratic, 431 histogram, 398 Inscribed
quotients of, 174 of hyperbolas and lines, 141 angles, 84
of inequalities, 370-372 polygons, 26
ranges of, 152-154 logarithmic equations, 419-420 Intercepts
rational, 665-669 of parabolas, 358-361, 430-434 x, 74
reciprocal, 162-163 of piecewise functions, 663 y, 74
of points, 1 Interior angles, 7
cosecant, 546 of quadratic functions, 308
cotangent, 548 of a polygon, 91
secant, 546 of rational functions, 665-667 Interquartile range, 403
sinusoidal, 375-379 regions of, 370-372 Intersection(s)
cosine, 375-376 of secant, 546-548
sine, 375-377 of sine, and cosine, 308-312 of lines, 2
sums of, 174 sinusoidal, 308 of planes, 3
tests for, 156-157 stem-and-leaf plot, 401 Inverse
horizontal line test, 157 of tangent, 548-549 functions, 219-222
vertical line test, 156 Greatest integer function, 663
trigonometric, 423, 508, 546-549 trigonometric, 222-231
powers of, 333 H cos"1,223-225
types of, 154-156 sin-1, 225-227
zeros of, 665 Half angle identities, 532-533 tan-1, 227-229
Fundamental Half lines. See Rays
counting principle, 270 Height, 9, 367. See also Altitude in logic, 58
theorem of arithmetic, 643 Histogram, 398 matrices, 658-660
Hyperbolas, 475-478, 596-598 Irrational
G numbers, 45, 77-78
asymptotes, 476, 477, 596-598 roots, 654-655
Gaussian distribution. See Normal as conic sections, 301 Isogonic triangles, 6
distribution equations of, 596-598, 671 Isosceles triangles, 7

General form standard form, 475, 671 Lateral surface area
of a circle, 302, 409 foci of, 475 definition of, 17
of a line, 282 locus definition of, 475 of right cylinders 17-19
Hypotenuse, 26, 27 of right prisms, 17-19
Geometric Hypotenuse-leg
figures (See also Polygons) congruency postulate, 69 Law of cosines, 489-493
area of, 45 Law of sines, 450-453
means, 535-537, 567-568 Identities.
progression, 535-537 double angle, 531-532, 553 ambiguous case, 557-559
common ratio, 535, 566 half angle, 532-533 proof of, 554-555
general expression, 565 product, 570-571 Least squares, 322
sequence, 588 (See also progression) sum and difference, 516-518, Leibniz, 458
571-573 Lengths
tangent, 518-519 of arcs, 281
of line segments, 2
L’Hopital, 458

745 Index

Limits, 198 Logic Numbers
Arguments in, 55-58 complex, 412-415
in trigonometric functions, 199 premises in, 55 composite, 643
reasoning in, 54 imaginary, 44-45
Linear syllogisms in, 55 irrational, 45, 77-78
statements of, 55-57 prime, 643
algebra, 458, 604 contrapositive, 57 rational, 77, 610
converse, 58 real, 44-45, 78
equations, 604 if-and-only-if, 58 relatively prime, 643
if-then, 56
forms of, 281-283 inverse, 58 O
negation of, 56-57
factor, 632 validity in, 55-56, 57 Oblique triangle, 6
Obtuse triangle, 6
regression, 243-244 M Odd functions, 465
Maclaurin, 458 One-to-one functions, 155, 157
with a calculator, 321-322 Matrix (Matrices) Open surfaces, 16
Origins
relationships, 406 addition, 603-608
additive identity, 604 of rays, 2
systems of equations algebra, 656 Outcomes, 499-500
definition of, 437 Outliers, 403
2X2, 577 determinant of, 438-439
elements, 437 P
3 X 3,576,577 minor, 591
multiplication, 605-606 Parabolas, 358-361, 383-385
n X n, 577, 578 order of, 603 as conic sections, 301
square matrices, 438 directrix of, 431, 432, 434
variation, 406 -407 zero determinate, 574-575 equation of, 359
Mean, 397-400, 441, 442 standard form, 671
velocity, 354 4 arithmetic, 513, 514, 567-568 focus of, 431,434
definition of, 397 graphs of, 431, 433, 434
Line segments geometric, 535-537, 567-568 locus definition of, 431
of the population, 441 translated, 433-434
bisecting, 38 Median vertex of, 431, 434
in statistics, 399, 441-442
congruence of, 2 of trapezoids, 237 Parallel lines, 7. See also Lines
of triangles, 116 conditions for, 8
copying, 37-38 Midpoint properties of, 8
formula method, 334-335
length of, 2 Minor, 591-592 Parallelograms, 233-237
of the element, 592 conditions for, 236-237
naming of, 1 of a matrix, 591 definition of, 233
Mode, 399, 441 properties of, 233-234
Lines Modulus, 286 types of, 234-236
Multiplication rule for equations, 124
equations of, 74-77 Multiplicative Pascal’s triangle, 472, 473, 580
identity, 604 Perimeter
locus definition of, 263 inverse, 657
Mutually exclusive events, 538 of circles, 9, 83
naming of, 1 Period
N
parallel, 3, 7 Nappes, 301 of a function, 423
Nonlinear of the sine function, 311
and planes, 3 of the tangent function, 548
systems of equations, 138-141 Periodic function, 308
skew, 3 Normal distribution, 398-400, 441 Permutations, 269-271
Notation circular, 363-364
slopes of, 74 conditional, 319-321
factorial, 193, 194 definition of, 461, 462
Local functional, 151, 174, 213 distinguishable, 364-365
permutation, 297, 298, 540 notation for, 297, 298, 540
maximum, 628 summation, 242, 243 Perpendicular, 4
bisectors, 334-335
minimum, 628 lines, 14, 76
Phase angle. See Phase shift
Locus, 263 Phase shift, 376-378, 423-425
Pi (7t)
definition of circles, 83, 301 numerical representation of, 77

definition of lines, 263

definition method, 334

definition of parabolas, 431

Logarithmic

equations, 184, 185, 387-390,

561-564

graphs of, 419-420

form,183, 184

functions, 338

graphs of, 419-420

inequalities, 618-619

Logarithm(s), 183-185, 387-390, 562

calculations with, 582-584

changing bases of, 560-561

common (base 10), 346-347

contrived problems for, 561-564

definition of, 389

natural (base e), 346-347

power rule for, 288-289

product rule for, 287

quotient rule for, 288

rules for, 287-289

development, 338-339

taking the, 495

746 Index

Piecewise functions, 662-663 Probability, 499-502 Quadrilaterals, 233-239
Planes, 3 of independent events, 501 ' categories of, 233
of mutually exclusive events, 538 definition of, 233
Argand (See complex) replacement, 501-502
complex, 412-413 simple, 499-500 Quotient rule
coordinate (See Coordinate systems, for logarithms, 288
Problems
rectangular) age, 178-179 Quotients
Point-slope form, 282 boat-in-the-river, 258-259 of functions, 174
Points, 1 clock, 510-511
Polar rate, 179-181 R.
abstract, 194-195,315-317
coordinates, 111, 413 overall average, 273 Radians
to rectangular, 111-112 abbreviation of, 277
form Product rule as angle measure, 277-281
for logarithms, 287 in circles, 277
of a complex number, 413, 551
Poles, 665 Products Radicals
Polygons of chord segments, 101-102 in equations, 50-51
of functions, 174 as exponents, 43
concave, 28, 90-91 of secant and tangent segments, in expressions, 43-44
convex, 28, 90-91 102-105 in trigonometric equations, 416
definition of, 28 as square roots, 43
diagonals of, 91-92 Progressions
exterior angles of, 91 arithmetic, 513, 535-537 Radii
interior angles of, 91 common difference of, 535 of circles, 9
regular (See Regular polygons) notation for, 536 forming angles, 3
Polynomial(s) geometric, 535-537 forming sectors, 9-10
complex, 644 common ratio, 535, 566
depressed, 625, 647 general expression, 565 Radius, 83. See also Circle(s)
division of, 124-125 Range
dominant term of, 629 Proofs
equations, 647 outlines for, 69-71 of a function, 152-154
functions, 630, 632 for overlapping triangles, 207-209 of the sine^ 311
general form of, 630 paragraph, 116 in statistics, 397
normalized equation, 638 of Pythagorean theorem, 127-128 Rate problems, 179-181
real, 644 of similarity, 128-130 abstract, 194-195,315-317
roots of, 681 two-column, 115-119 overall average, 273
turning points, 628 Rational
zeros, 625-626, 628, 630 Properties functions, 665-669
Population, 441 of isosceles trapezoids, 238
mean of the, 441 of parallelograms, 233 asymptopes of, 666
Postulate(s) of rectangles, 235 numbers, 77, 610
AAAS congruency, 68 of rhombuses, 235 roots theorem, 644-645
congruency, 68-69 of squares, 235 Rationalization
Euclid’s, 115 of trapezoids, 237 of denominators, 44, 78
HL congruency, 69 Ratios
SAS Proportionality of arc length to radius, 3
of segments, 62-63 Rays
congruency, 68 in angles, 2, 3
similarity, 129, 130 Protractor, 36 definition of, 2
SSS Pyramids naming of, 2
congruency, 14, 68 opposite, 2
similarity, 130 definition of, 19 Real
triangle inequality, 27, 543 surface area of, 21, 22 polynomials, 644
Power of the hydrogen, 584 volume of, 19 Rectangles
Power rule Pythagoras, 26 areas of, 9, 14
for logarithms, 288-289 Pythagorean Rectangular
Powers, 166 inequalities, 27 coordinates, 110-111,412
definition of, 495 proof of, 127-128
of trigonometric functions, 333 theorem, 26-27, 46, 47 to polar, 111-112, 205-207
Primitive terms, 1 form,112'
Principle square root, 483 Quadrants, 189 solids, 43,46-47
Prisms signs in, 222
definition of, 16 diagonals of, 46-47
lateral surface area of, 17, 18, 20, 21 Quadratic Reference triangles
volume of, 17 equations
with designated roots, 272 with degrees, 146
formula, 87-88, 139 with radians, 294, 295, 296
Reflections, 211, 213
Region of interest, 638
Regions, 3
of graphs, 370-372

Regular polygons, 21, 455, 456 Similarity Symbols (cont.)
in finding area, 45
apothem, 455 of polygons, 28-29 JR (real numbers), 152
in triangles, 29-30, 61
area of, 456 SF (scale factor), 30
Simple probability, 499, 500 {) (set notation), 152
perimeter of, 455 events, 499 ~ (similarity), 61
experiments, 499 <7 (standard deviation), 397, 441
Related angels, 188 outcomes, 499 S (summation), 242
sample space, 499 U (union), 539
Relations, 155, 156 Symmetry
Sine, 468, 489, 491,516,570,571 axis of, 212
Relationships functions, 308 point of, 211
graph, 311 Synthetic division, 624-626
linear, 406 as ratio of sides, 108, 197-198 Systems
2x2 systems, 577
Relatively prime numbers, 643 Single-variable data 3X3 systems, 576, 577
definition of, 397 n X n systems, 577, 578
Remainder theorem, 636, 639 of equations, 422-428, 680-681
Sinusoids, 375-378, 424
Resultant, 205 amplitude, 375-378, 424 two linear, 50, 405, 458 *
graphs of, 309-312 three linear, 51-52
Right period, 375-378, 423, 425 nonlinear, 138-141
phase angle (See phase shift) independent, 577-578
cylinders, 16-18 phase shift, 376-378, 423-425 of inequalities, 370-372
sketching, 505, 506
triangles, 6, 450 translations of, 328-330 Tangent. See also Circle(s)
to a circle, 83
solids, 16, 19 Skew lines, 3 identities, 518-519
Slope-intercept form, 282 lines, 83, 97
prisms, 16, 17 Slopes. See Lines period of, 548
Solutions. See Roots as ratio of sides, 108
cones, 19-21 # Sorenson, 584 as sine and cosine, 198
Spheres, 22-24
angles (See Angle(s)) Square roots. See Radicals Terminal side, 4, 491
Side-side-side Tests
Roots
congruency postulate, 14, 68 horizontal line, 157
complex, 324-325, 551 similarity postulate, 130 vertical line, 156
Standard deviation, 397-400, 441, 442 Theorem(s)
of complex numbers, 481-483, 551 definition of, 397 of arithmetic, fundamental, 643
Standard form, 44-45 binomial, 621
of equations, 625, 644 of a circle, 302, 409, 671 chord-tangent, 93-94
of an ellipse, 446, 475, 526, 671 definition of, 115
irrational, 654-655 of a hyperbola, 475, 671 rational roots, 644, 645
of a parabola, 671 remainder, 637, 639
of polynomial equations, 647 Stem-and-leaf plots, 401 turning-point, 628
Summation upper and lower bound, 652-654
of polynomials, 681 notation, 242-243 zero-factor, 630
variable of, 242 Third quartile, 402
Rotations Sum(s) Tick marks
of functions, 172-174 to designate congruence, 5
of rays to form angles, 4 of trigonometric functions, 256-258 Translation, 214-216
of two cubes, 142-143 of conic sections, 595
S Surface areas of parabolas, 433-434
of cones, 20, 21 of sinusoids, 328-330
Sample space, 499, 500 of cylinders and prisms, 17-19 Transversals, 7
Side-angle-side of spheres, 22-24 cutting equal ratios, 9
Symbols Trapezoids, 237-239
congruency postulate, 68 Z (angle), 4 altitude of, 237
similarity postulate, 129, 130 CO (angular velocity), 354 area of, 238, 368
Scalar product, 604 = (congruency), 2, 69 definition of, 237
Scale factor, 29, 45, 46 0 (degree), 4 isosceles, 238
symbol, 30 e (element of), 152 median of, 237
Scalene triangle, 7 i (imaginary; a/—T), 44-45 properties of, 237
Secants, 293, 469. See also Circle(s)
definition of, 83, 97 (intersection), 539
intersecting, 97-100, 101 ~ (inverse), 58
segments, 102 x (mean), 397, 441

external, 102
Sectors

definition of, 9
Segments

of circles, 369
of lines, 1 (See also Line segments)
Semicircles, 83
Sequence, 512-513, 514, 515
arithmetic, 513-515, 586, 587 (See

also Arithmetic, progressions)

finite, 512
geometric (See Geometric,

progression)
infinite, 512
notations, 565
Series
arithmetic, 586-587
definition of, 586
geometric, 586, 588-589, 600

748 Index

Triangle(s) Trigonometric, functions (cont.) Vectors
altitude, 9 power of, 333 addition of, 205-207
areas of, 9, 14-15, 554 reciprocal, 293-296 in polar form, 205-207
constructing, 39 signs of, 188-191 in rectangular form, 205-207
degrees in, 6 sums of, 172
inequality postulate, 27, 543 types of Velocity
median, 116 cosecant, 546-548 angular, 354-356
overlapping, 207 cosine, 309-312 linear, 354
Pascal’s, 472, 473, 580 cotangent, 548-549
reference, 146-148 secant, 546-548 Vertex (Vertices)
types of, 6, 7 sine, 309-312 of angles, 4
30°-60°-90°, 146 tangent, 548-549 of an ellipse, 447
45°-45°-90°,146 values of, 145, 146, 199-201 of a hyperbola, 476, 477, 478
acute, 6 of a parabola, 431, 434
equiangular, 6 identities, 468-470, 486, 517, 542 of a polygon, 29
equilateral, 6 ratios, 145
isogonic, 6 Trinomial, 79, 80 Volume(s)
isosceles, 7 Turning point(s), 628 of cones, 19
oblique, 6, 450 theorem, 628 of cylinders, 17, 23
obtuse, 6 Two-point form, 282 of spheres, 22, 23
right, 6, 450
scalene, 7 U W
similar, 61, 555 Undefined terms, 2. See also Primitive Word problems

Triangulate, 91 terms advanced, 133-136
Trigonometric Unit(s)
X
equations, 342-343, 392-394, of angular velocity, 354 x coordinates, 110
508-509 circle, 197, 198, 309 x intercept, 74
multipliers, 353-354
arguments in, 350-351 Upper and lower bound theorem, Y>
radicals in, 416 y coordinates, 110
expressions 652-654 y intercept, 74
factoring, 392-394, 504-505
functions V Z
absolute value, 188, 258 Variables, 405 Zero(s)
addition of, 256-258
graphing of dependent, 151, 419, 423 determinate, 574-575
independent, 150, 419, 423 factor theorem, 272, 630
cosine, 309-312 of summation, 242 of functions, 665
sine, 309-312 Variance, 397 of a polynomial, 625-626, 628
tangent, 548-549



'-

»









y. >
1

Co 3 1127 00D52255 4

DATE DUE
DEC l 1 « W

DEC 1 9 i 998

rU\brt1 i| 0n tAuAuA4R

GAYLORD PRINTED IN U.S. A.

5 12
Sa9
g • 11 -
12
1996

_4

MarygrovQ College Library
Curriculum Center

Detroit, Michigan 48221