I-13 INDEX Unit fractions, 614 Unit vector, 813 Universal set, 16 Unlike radicals, 87 Upper harmonics, 747, 752 V Value absolute, 24 – 25, 180 – 184 future, 109, 455, 1066–1067 maturity, 109 present, 455 Variable dependent, 221, 223 independent, 221, 223 solving for a specified, 109, 137 Variation combined, 413 constant of, 411 direct, 411 directly as the nth power, 412 inverse, 413 inversely as the nth power, 413 joint, 413 in sign, 354 steps to solve problems with, 412 Vector(s) algebraic interpretation of, 811 – 812 angle between, 816 applications of, 804–806 components of, 811 direction angle for, 811 dot product of, 814–816 equilibrant of, 803 horizontal component of, 811 i, j units, 814 initial point of, 801 inner product of, 814 magnitude of, 801 naming practices, 801 operations with, 813 opposite of, 802 orthogonal, 816 parallelogram rule for, 802 position, 811 quantities, 801 resultant of, 802, 821 symbol for, 801 terminal point of, 801 unit, 813 vertical component of, 811 zero, 802 Vector cross products, 908 Velocity constant, 113 in motion problems, 113 non-constant, 113 Venn diagram, 17, 1101 Verifying trigonometric identities, 689 – 693, 706 Vertex (Vertices) of an ellipse, 1002 of a hyperbola, 1015 of a parabola, 269, 325, 329, 992 of a polygonal region, 944 solving by squaring, 743 solving using the quadratic formula, 742 solving by zero-factor property, 741 Trigonometric form of a complex number, 822 Trigonometric function(s) circular, 607 cofunctions of, 550, 699–700 combinations of translations of, 637 definitions of, 534 derivatives of, in calculus, 689 domains of, 609 interpreting in calculus, 213 inverses of, 724–731 ranges of, 542 right-triangle-based definitions of, 549 translations of, 633–638 Trigonometric function values of acute angles, 549 finding with a calculator, 556 – 557 of nonquadrantal angles, 555 of quadrantal angles, 537–538 signs and ranges of, 540 of special angles, 551–552 undefined, 538 Trigonometric identities cofunction, 550, 699–700 difference, 697 – 698, 700 – 701, 702 – 705 double-angle, 711 – 714 even-odd, 682 – 683 fundamental, 682 – 683 half-angle, 717 – 720 product-to-sum, 715 – 716 Pythagorean, 543–544, 683 quotient, 544, 683 reciprocal, 539, 683 solving conditional trigonometric equations using, 742–743 sum, 697 – 698, 700 – 701, 702 – 704 sum-to-product, 716 – 717 verifying, 689–693, 706 Trigonometric models, 628, 638 – 639 Trigonometric substitution, 544 Trinomials definition of, 51 factoring, 63 – 65 perfect square, 54, 64–65 Triples, ordered, 204, 880 Trochoid, 854 Turning points of polynomial function graphs, 362 U Undefined slope, 241–242, 243 Union of sets, 18, 19, 173, 1100 Union of two events definition of, 1100 probability of, 1102–1103 Unit, imaginary, 123 Unit circle, 213, 607 difference identity for, 700–701 domain of, 645 double-angle identity for, 712 graph of, 645 half-angle identity for, 718 inverse of, 729–730 period of, 645 range of, 542, 645 steps to graph, 647 sum identity for, 700–701 vertical translation of, 649 Tangent line to a circle, 219 to a curve, 253 Tartaglia, Niccolo, 358 Term(s) coefficient of, 51 definition of, 51 dominating, 324, 362 like, 51 of a polynomial, 51 of a sequence, 1040 Terminal point of a vector, 801 Terminal side of an angle, 526 Terminating decimals, 9 Tests for symmetry, 282–284 Three-part inequalities, 170–171 Threshold sound, 479 Tolerance, 184 Traffic intensity, 392 Transformations of linear systems, 875 Transit, 576 Translation(s) of an ellipse, 1005–1006 combinations of, 637–638 of graphs, 286, 287, 360 horizontal, 287, 633–635, 649 of a hyperbola, 1018 of a parabola, 996 summary of, 290 vertical, 286, 636, 649 Transverse axis of a hyperbola, 1015 Tree diagram, 1088 Trial, 1099 Triangle(s) acute, 773 area of, 780, 791 congruent, 772 Heron, 799 oblique, 772 – 775 obtuse, 773 Pascal’s, 1073 – 1074 perfect, 799 right. See Right triangle(s) solving, 567 – 568, 774 – 775, 777 – 779, 789 – 791 Triangulation method, 798 Trigonometric equations conditional, 740 with half-angles, 744 inverse, 753 – 756 linear methods for solving, 740 with multiple angles, 745–746 quadratic methods for solving, 741 – 742
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