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26_169841 bindex.qxd 2/29/08 2:58 PM Page 365 Index addition law, for limits, 345 Symbols and Numerics additive identity, 14 > (greater than), 21 additive inverse property, 14 ≥ (greater than or equal to), 21 advanced identities, described, 195 < (less than) symbol, 21 Algebra, Fundamental Theorem of, 82–83, 95 ≤ (less than or equal to), 21 algebra, labeling and expressing ellipses, θ (theta) 269–270 graphing calculator, entering as a variable algebra I and II, compared to pre-calculus, 10 in, 251–252 Algebra I and II Workbook For Dummies as the input of a trig function, 147 (Sterling), 240, 356 length of radius, 243 Algebra II For Dummies (Sterling), 240 as plotting point, 244 algebraic expressions, simplifying, 178 theta prime, as the name given to the algebraic method, to find limit, 342–345 reference angle, 141 amplitude in trig functions, 125 changing, 160–163, 168–169 2 x 2 matrix, inverse of, 310 defined, 160 2(π) radian, 150 period, compared to, 165 3 x 3 system of equations, and Cramer’s positive values of, 161 rule, 314 sine functions, 161–163 30-60 triangle, 132–133, 135 transformations, combining, 167–170 45er triangle, 131–132, 138 analytical method for finding a limit, 360, convenience of, 120 341–342 AND statement, 25 angle(s) • A • arc as, 145 central, 145–146 AAS (Angle Angle Side) triangle, solving, common, 128–129 218, 220–221 correct placement, 134–136 absolute value, 13, 22–24, 36, 48 cosine of, 122–123 absolute-value functions, 36 in degrees, 197 acceleration, moving objects with, 338 doubling the trig value of, 205–207 acute triangle, 222 COPYRIGHTEDdrawing, MATERIAL 120, 130–131 addition formulas for finding, 200–202, 229 associative property of, 13 knowing two measures, 219–221 commutative property of, 13 negative, 130–131, 246 of complex numbers, 241 reference angles, 141–144 of degrees, 197 sides, using to find, 228–230 of exponents, to multiply, 362 sine of, 121–122, 139, 197, 227 of fractions, 11, 179 as sums or differences, 196–197, 200–202 functions, 58–59 tangent of, 123–124, 203 of matrices, 297–298 on unit circle, 129, 140–144 of radians, 197 26_169841 bindex.qxd 2/29/08 2:58 PM Page 366 366 Pre-Calculus For Dummies Angle Angle Side (AAS) triangle, solving, base, 103, 106 218, 220–221 base numbers, fractions as, 99 Angle Side Angle (ASA) triangle, solving, binomial coefficients, 329–336 218–220 binomial equation, factoring, 69 angle theta, deriving from the tangent, 248 binomial expansion, 329, 332–336 Annual Percentage Rate (APR), 321–322 binomial theorem answer matrix, 301 breaking down into parts, 330 answers, checking, 284, 355 defined, 329–330 antilogarithm, 106 expanding by using, 332–336 APR (Annual Percentage Rate), 321–322 using, 329 arc length, calculating, 145–146 binomials, 67, 74, 329 arccos (inverse cosine), 125 biology, exponential concepts in, 98 Archimedes Spiral, graphing, 250 British Method, 30, 69, 71–73, 241 arcs, making and measuring, 145–146 arcsin (inverse sine), 125, 223 arctan (inverse tangent), 125–126 • C • area, calculating triangle, 232–233 calculator, and order of operations, 231. Argand coordinate plane, 238 See also graphing calculator arithmetic sequences, 319–321, 325–326 calculus Arone, Wendy (author), 228 compared to pre-calculus, 338–339 ASA (Angle Side Angle) triangle, solving, described, 337 218–220 habits helping, 351–357 associative property of habits to break before, 359–363 addition/multiplication, 13 cancellation, 362–363 asymptote Cardiod, graphing, 250 changing for logs, 109 Cartesian coordinates, 15, 247, 278–282 of cosecant graph, 158 center of cotangent graphs, 152 of ellipse, 269–270 of a graph, 49 of hyperbola, 274–276 graphing, 53–54 center-radius form, 257 horizontal, 51, 53–56, 98, 101–102 central angle, 145–146 of hyperbola, 275–277 change of base formula, 105 oblique, 51–52 checking, answers, 284, 355 of secant graph, 156 circle(s) of tangent graphs, 152 circumference formula, 145 vertical, 49–50, 53–57, 153, 155 described, 254–255, 257 attack, planning, 352–353 eccentricity value for, 281 augmented form, of a matrix, 305 graphing, 250, 257–259 axes, of ellipses, 271 unique characteristics of, 256 axis of symmetry, 260, 263–266, 274 closed interval, 25 co-function identities, 185–187, 202, 204 co-terminal angles, 120 • B • co-vertices of ellipse, 269–272 back-solving, executing, 304–305 coefficient matrix, 301, 311–313 back-substitution process, described, 293 coefficients, 39 bad math habits, breaking, 359–363 coefficients of binomials, 331–332 26_169841 bindex.qxd 2/29/08 2:58 PM Page 367 Index 367 combination formula, using, 334 intersection of, 285 combinations, calculator button for, 331 no more than four solutions for two, 290 combined functions, 58–62 parabolas as, 260 common denominator, for angles in parametric form, 278–280 radians, 198–199 on polar plane, 280–282 common difference, 319–321 conics, expressed outside of Cartesian common logarithms, 103, 111 coordinates, 278–282 common ratio, finding, 326 conjugate, multiplying by, 30, 191–192, commutative property 204, 344 of addition, 13 conjugate axis, of a hyperbola, 274, 276 of multiplication, 13, 212 consecutive terms, 319–320, 322–323 comparisons, of expressions, 16 consistent and independent system, 285 completing the square, described, 78–80 constants complex conjugates, 82, 241 cannot be added to variables, 360 complex coordinate plane, 238, 242 in conic equations, 256 complex numbers described, 68, 294 adding, 241 multiplying equations by, 288 defined, 12, 238 scalar, 241, 297–298 dividing, 241–242 continuity, in functions, 346–348 expansion with, 335–336 continuous graph, 346 expressing answers as, 95 conversion factor, for radians, 247–248 form of, 83 coordinate pair (x, y), 15, 246 graphing, 238, 242–243 coordinate plane manipulating arithmetically, 240 described, 33 multiplying, 240–241 graphing on, 16 operations involving, 240–242 points existing on, 126 with real and imaginary parts, 240 shifting the waves on, 166–167 subtracting, 241 unit circle lying on, 126 system of, 83, 239–242 coordinates, of points on unit circle, 135 types of, 240 cosecant complex roots, 83, 240 of angle, 139 complicated polar coordinates, 246 defined, 124 composed function, 60–62 evaluating on the unit circle, 138 composition of functions, breaking down, 60 graphing, 156, 158–159, 173–176 computer program, finding matrix cosine inverses, 310 of angle, 122–123, 139 concepts, understanding, 356 changing functions in equation to, 180 cones, cutting with a plane, 254–255 defined, 122 congruence postulates, for triangles, 228 double angle of, 207–208 congruent angles, same values for different half-angle formula, 210 trig functions, 134 point-in-the-plane definition, 137 conic sections reciprocal of, 124 characteristics unique to each type of, 256 sine, changing to, 208 described, 253 sum and difference formulas, 200–202 equations of, 256–257 cosine ⋅ cosine product-to-sum graphing, 278–282 formula, 212 identifying, 254–257 cosine functions, 148–151, 170, 174 26_169841 bindex.qxd 2/29/08 2:58 PM Page 368 368 Pre-Calculus For Dummies cosine graphs difference cosine parent graph, characteristics cosine, formula for, 200 of, 151 of cubes, 74, 76 described, 150–151 sine, formula for, 197 springs, compared to, 160–170 of squares, 74–75 tangent and cotangent, compared to, 152 tangent, formula for, 203–205 cotangent, 125, 138–139, 152 transporting from sums or differences to, cotangent functions, graphing, 152, 154–155 213–214 cotangent graphs, transforming, 170–173 using formulas, 198–199 counting numbers, defined, 11 difference-to-product identities, 213 Cramer, Gabriel (mathematician), 311 dimensions, of a matrix, 297 Cramer’s rule, 311–314 directrix for parabolas, 260, 263–266 critical point, 35, 92–93 Dirty Monkeys Smell Bad mnemonic, 86 cross product, 299 discontinuity, dealing with, 347–348 cube root, 27, 29, 37–38 discontinuous function, 48–49, 337 cubic functions, 37, 42–43 discriminant, 82, 240 curve, area under, 338 disjoint sets, 25 distance, formula for, 17, 127 distributing, improper, 363 • D • distributive property, 14, 197 decay constant, 113 dividend, defined, 85–86 decomposing, partial fractions, 293–295 division degree mode, setting calculator to, 123 of functions, 59 degrees of polynomials, 51 adding and subtracting, 197 rational roots, 86–88 calculating in, 197–198 synthetic, 88–90, 96 of numerator and denominator, 52–53 division algorithm, 86 of polynomials, 51, 68, 81 division bar, as grouping symbol, 360 working with angles in, 196 division law, for limits, 345 denominators divisor, defined, 85–86 conjugate of, 30 DNE value, 343 containing a binomial with a radical, 30 domain equal degree with numerators, 55–56 combined function, 61 with greater degree, 53–55 cosecant graph, 176 multiplying by conjugates of, 192 cosine function, 150, 170 rationalizing, 28–31, 204 cotangent function, 173 splitting up, 360 exponential function, 99 in trig proofs, 189–192 of relation, 15 dependent system, 285 secant function, 175 depreciation, 321–322, 324, 326 sine function, 148 depressed polynomial, 84, 90, 96 tangent function, 172 derivative, finding, 346 x value as, 147 Descartes, Rene (mathematician), 15 double-angle formula, 205–209 Descartes’ Rule of Signs theorem, double root, 89, 95 81, 83–84, 95 drawing determinants of coefficient matrices, while problem solving, 352 311–314 to represent right triangles, 201 for trig problems, 130 26_169841 bindex.qxd 2/29/08 2:58 PM Page 369 Index 369 even functions, 34, 151, 183 • E • even multiplicity, at a root, 95 eccentricity value, conic

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