(BQ) Part 2 book College algebra trigonometry has contents: The circular functions and their graphs, trigonometric identities and equations, applications of trigonometry, analytic geometry, further topics in algebra.
www.downloadslide.com The Circular Functions and Their Graphs Phenomena that repeat in a regular pattern, such as average monthly temperature, fractional part of the moon’s illumination, and high and low tides, can be modeled by periodic functions 6.1 Radian Measure 6.2 The Unit Circle and Circular Functions 6.3 Graphs of the Sine and Cosine Functions 6.4 Translations of the Graphs of the Sine and Cosine Functions Chapter Quiz 6.5 Graphs of the Tangent and Cotangent Functions 6.6 Graphs of the Secant and Cosecant Functions Summary Exercises on Graphing Circular Functions 6.7 Harmonic Motion 591 M07_LIAL1953_06_GE_C06.indd 591 31/08/16 4:53 pm www.downloadslide.com 592 CHAPTER 6 The Circular Functions and Their Graphs 6.1 Radian Measure ■ Radian Measure Radian Measure We have seen that angles can be measured in degrees In more theoretical work in mathematics, radian measure of angles is preferred Radian measure enables us to treat the trigonometric functions as functions with domains of real numbers, rather than angles Figure shows an angle u in standard position, along with a circle of radius r The vertex of u is at the center of the circle Because angle u intercepts an arc on the circle equal in length to the radius of the circle, we say that angle u has a measure of radian ■ Conversions between Degrees and Radians ■ Arc Length on a Circle ■ Area of a Sector of a Circle y Radian r U r u = radian Figure x An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of radian It follows that an angle of measure radians intercepts an arc equal in length to twice the radius of the circle, an angle of measure radian intercepts an arc equal in length to half the radius of the circle, and so on In general, if U is a central angle of a circle of radius r, and U intercepts an arc of length s, s then the radian measure of U is r See Figure y y y 2r U x r C = 2Pr u = radians U r r u= radian U x r x u = 2p radians Figure s The ratio r is a pure number, where s and r are expressed in the same units Thus, “radians” is not a unit of measure like feet or centimeters Conversions between Degrees and Radians The circumference of a circle—the distance around the circle—is given by C = 2pr, where r is the radius of the circle The formula C = 2pr shows that the radius can be measured off 2p times around a circle Therefore, an angle of 360°, which corresponds to a complete circle, intercepts an arc equal in length to 2p times the radius of the circle Thus, an angle of 360° has a measure of 2p radians 360° = 2P radians An angle of 180° is half the size of an angle of 360°, so an angle of 180° has half the radian measure of an angle of 360° 180° = M07_LIAL1953_06_GE_C06.indd 592 1 2P2 radians = P radians Degree/radian relationship 31/08/16 4:53 pm www.downloadslide.com 6.1 Radian Measure 593 We can use the relationship 180° = p radians to develop a method for converting between degrees and radians as follows 180° = P radians Degree/radian relationship 1° = P 180° radian Divide by 180. or 1 radian = Divide by p P 180 NOTE Replacing p with its approximate integer value in the fractions above and simplifying gives a couple of facts to help recall the relationship between degrees and radians Remember that these are only approximations 1° ≈ radian and radian ≈ 60° 60 Converting between Degrees and Radians p • Multiply a degree measure by 180 radian and simplify to convert to radians • Multiply a radian measure by EXAMPLE 180° p and simplify to convert to degrees Converting Degrees to Radians Convert each degree measure to radians (a) 45° (b) - 270° (c) 249.8° SOLUTION (a) 45° = 45a This radian mode screen shows TI-84 Plus conversions for Example Verify that the first two results are approximations for the exact values p 3p of and - p p p radian b = radian Multiply by 180 radian 180 (b) - 270° = -270 a (c) 249.8° = 249.8 a EXAMPLE p p 3p Multiply by 180 radian radian b = radians Write in lowest terms 180 p radian b ≈ 4.360 radians Nearest thousandth 180 ■ ✔ Now Try Exercises 11, 17, and 45 Converting Radians to Degrees Convert each radian measure to degrees (a) 9p 5p (b) (c) 4.25 SOLUTION (a) 9p 9p 180° radians = a b = 405° Multiply by 180° p p 4 (b) This degree mode screen shows how a TI-84 Plus calculator converts the radian measures in Example to degree measures M07_LIAL1953_06_GE_C06.indd 593 5p 5p 180° radians = a b = -150° Multiply by 180° p p 6 (c) 4.25 radians = 4.25a 180° b ≈ 243.5°, or 243° 30′ 0.50706160′2 ≈ 30′ p ■ ✔ Now Try Exercises 29, 33, and 57 31/08/16 4:53 pm www.downloadslide.com 594 CHAPTER 6 The Circular Functions and Their Graphs NOTE Another way to convert a radian measure that is a rational multiple 9p of p, such as , to degrees is to substitute 180° for p In Example 2(a), doing this would give the following 91180°2 9p radians = = 405° 4 One of the most important facts to remember when working with angles and their measures is summarized in the following statement Agreement on Angle Measurement Units If no unit of angle measure is specified, then the angle is understood to be measured in radians For example, Figure 3(a) shows an angle of 30°, and Figure 3(b) shows an angle of 30 (which means 30 radians) An angle with measure 30 radians is coterminal with an angle of approximately 279° y y 30s x x 30 degrees 30 radians (a) (b) Note the difference between an angle of 30 degrees and an angle of 30 radians Figure The following table and Figure on the next page give some equivalent angle measures in degrees and radians Keep in mind that 180° = P radians Equivalent Angle Measures Degrees Radians Degrees Exact Approximate 0° 0 90° 30° p 0.52 45° p 60° p Radians Exact p Approximate 180° p 3.14 0.79 270° 3p 4.71 1.05 360° 2p 6.28 1.57 These exact values are rational multiples of p M07_LIAL1953_06_GE_C06.indd 594 31/08/16 4:53 pm www.downloadslide.com 6.1 Radian Measure 595 y LOOKING AHEAD TO CALCULUS In calculus, radian measure is much easier to work with than degree measure If x is measured in radians, then the derivative of ƒ1x2 = sin x is ƒ′1x2 = cos x However, if x is measured in degrees, then the derivative of ƒ1x2 = sin x is ƒ′1x2 = p cos x 180 120° = 135° = 3P 5P 150° = 2P 90° = P 60° = P P 45° = P 30° = 180° = P 0° = 7P 225° = 5P 240° = 4P 210° = 270° = 3P x 330° = 11P 315° = 7P 300° = 5P Figure Learn the equivalences in Figure They appear often in trigonometry Arc Length on a Circle The formula for finding the length of an arc of a s circle follows directly from the definition of an angle u in radians, where u = r In Figure 5, we see that angle QOP has meay sure radian and intercepts an arc of length r on T Q the circle We also see that angle ROT has measure u radians and intercepts an arc of length s on s r the circle From plane geometry, we know that the U radians radian lengths of the arcs are proportional to the meax r O r R P sures of their central angles s u = Set up a proportion r Figure Multiplying each side by r gives s = r u. Solve for s Arc Length The length s of the arc intercepted on a circle of radius r by a central angle of measure u radians is given by the product of the radius and the radian measure of the angle s = r U, where U is in radians CAUTION When the formula s = rU is applied, the value of U MUST be expressed in radians, not degrees M07_LIAL1953_06_GE_C06.indd 595 31/08/16 4:53 pm www.downloadslide.com 596 CHAPTER 6 The Circular Functions and Their Graphs EXAMPLE Finding Arc Length Using s = r U A circle has radius 18.20 cm Find the length of the arc intercepted by a central angle having each of the following measures (a) 3p radians (b) 144° SOLUTION (a) As shown in Figure 6, r = 18.20 cm and u = s = r u 3p Arc length formula 3p s = 18.20 a b Let r = 18.20 and u = s ≈ 21.44 cm 3P s r = 18.20 cm 3p Use a calculator Figure (b) The formula s = r u requires that u be measured in radians First, convert u p to radians by multiplying 144° by 180 radian 144° = 144 a p 4p b = radians Convert from degrees to radians 180 The length s is found using s = r u s = r u = 18.20 a Be sure to use radians for u in s = r u 4p b ≈ 45.74 cm Let r = 18.20 and u = 4p ■ ✔ Now Try Exercises 67 and 71 Latitude gives the measure of a central angle with vertex at Earth’s center whose initial side goes through the equator and whose terminal side goes through the given location As an example, see Figure EXAMPLE Finding the Distance between Two Cities Reno, Nevada, is approximately due north of Los Angeles The latitude of Reno is 40° N, and that of Los Angeles is 34° N (The N in 34° N means north of the equator.) The radius of Earth is 6400 km Find the north-south distance between the two cities SOLUTION As shown in Figure 7, the central angle between Reno and Los Reno s 6° Los Angeles 6400 km Figure 40° - 34° = 6° The distance between the two cities can be found using the formula s = r u, after 6° is converted to radians 40° 34° Angeles is Equator 6° = a The distance between the two cities is given by s s = r u = 6400 a M07_LIAL1953_06_GE_C06.indd 596 p p b = radian 180 30 p p b ≈ 670 km Let r = 6400 and u = 30 30 ■ ✔ Now Try Exercise 75 31/08/16 4:53 pm www.downloadslide.com 6.1 Radian Measure EXAMPLE 39.72° 597 Finding a Length Using s = r U A rope is being wound around a drum with radius 0.8725 ft (See Figure 8.) How much rope will be wound around the drum if the drum is rotated through an angle of 39.72°? 0.8725 ft SOLUTION The length of rope wound around the drum is the arc length for a circle of radius 0.8725 ft and a central angle of 39.72° Use the formula s = r u, with the angle converted to radian measure The length of the rope wound around the drum is approximated by s Figure Convert to radian measure s = r u = 0.8725c 39.72 a EXAMPLE p b d ≈ 0.6049 ft 180 ■ ✔ Now Try Exercise 87(a) Finding an Angle Measure Using s = r U Two gears are adjusted so that the smaller gear drives the larger one, as shown in Figure If the smaller gear rotates through an angle of 225°, through how many degrees will the larger gear rotate? m 5c 4.8 cm Figure SOLUTION First find the radian measure of the angle of rotation for the smaller gear, and then find the arc length on the smaller gear This arc length will correspond to the arc length of the motion of the larger gear Because 5p 225° = radians, for the smaller gear we have arc length s = r u = 2.5 a 5p 12.5p 25p b = = cm 4 The tips of the two mating gear teeth must move at the same linear speed, or the teeth will break So we must have “equal arc lengths in equal times.” An arc with this length s on the larger gear corresponds to an angle measure u, in radians, where s = r u s = r u Arc length formula 25p = 4.8u Let s = 25p and r = 4.8 (for the larger gear) 125p =u 192 4.8 = 48 10 = 24 ; Multiply by 24 to solve for u Converting u back to degrees shows that the larger gear rotates through 125p 180° a b ≈ 117°. Convert u = 125p 192 to degrees p 192 ■ ✔ Now Try Exercise 81 U r Figure 10 M07_LIAL1953_06_GE_C06.indd 597 The shaded region is a sector of the circle Area of a Sector of a Circle A sector of a circle is the portion of the interior of a circle intercepted by a central angle Think of it as a “piece of pie.” See Figure 10 A complete circle can be thought of as an angle with measure 2p radians If a central angle for a sector has measure u radians, then the sector u makes up the fraction 2p of a complete circle The area 𝒜 of a complete circle with radius r is 𝒜 = pr Therefore, we have the following Area 𝒜 of a sector = u 1pr 22 = r u, where u is in radians 2p 31/08/16 4:54 pm www.downloadslide.com 598 CHAPTER 6 The Circular Functions and Their Graphs Area of a Sector The area 𝒜 of a sector of a circle of radius r and central angle u is given by the following formula 𝒜= r U, where U is in radians CAUTION As in the formula for arc length, the value of U must be in radians when this formula is used to find the area of a sector EXAMPLE Finding the Area of a Sector-Shaped Field m A center-pivot irrigation system provides water to a sector-shaped field with the measures shown in Figure 11 Find the area of the field Center-pivot irrigation system 15° = 15 a p p b = radian Convert to radians 180 12 Now find the area of a sector of a circle 𝒜= 𝒜= Figure 11 r u Formula for area of a sector p 132122 a b Let r = 321 and u = 12 𝒜 ≈ 13,500 m2 6.1 15° 321 SOLUTION First, convert 15° to radians p 12 Multiply. ■ ✔ Now Try Exercise 109 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence 1 An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the of the circle has measure radian 360° = radians, and 180° = radians 3 To convert to radians, multiply a degree measure by radian and simplify 4 To convert to degrees, multiply a radian measure by and simplify CONCEPT PREVIEW Work each problem 5 Find the exact length of the arc intercepted by the given central angle 6 Find the radius of the circle 6P 3P P M07_LIAL1953_06_GE_C06.indd 598 31/08/16 4:54 pm www.downloadslide.com 6.1 Radian Measure Find the measure of the central angle (in radians) 599 Find the area of the sector 15P 20 U U 10 10 Find the measure (in radians) of the central angle The number inside the sector is the area 10 Find the measure (in degrees) of the central angle The number inside the sector is the area 96P sq units 12 sq units Convert each degree measure to radians Leave answers as multiples of p See Examples 1(a) and 1(b) 11 300° 12 225° 13 240° 14 45° 15 315° 16 2250° 17 -90° 18 - 270° 19 690° 20 675° 21 2025° 22 1230° 23 135° 24 -740° 25 -800° 26 - 610° Convert each radian measure to degrees See Examples 2(a) and 2(b) 27 p 28 4p 29 5p 31 7p 32 15p 33 - 5p 34 - 7p 35 21p 20 36 31p 20 37 - 17p 10 38 - 13p 10 39 17p 20 40 11p 30 41 -12p 30 2p 42 - 9p Convert each degree measure to radians If applicable, round to the nearest thousandth See Example 1(c) 43 23° 44 74° 45 42.5° 46 264.9° 47 144° 50′ 48 174° 50′ 49 81.91° 50 85.04° 51 56° 25′ 52 122° 37′ 53 -53.91° 54 - 23.01° Convert each radian measure to degrees Write answers to the nearest minute See Example 2(c) 55 56 57 4.48 58 3.06 59 1.6684 60 0.1194 61 -4.95972 62 - 2.26678 63 Concept Check The value of sin 30 is not Why is this true? 64 Concept Check What is meant by an angle of one radian? M07_LIAL1953_06_GE_C06.indd 599 31/08/16 4:54 pm www.downloadslide.com 600 CHAPTER 6 The Circular Functions and Their Graphs 65 Concept Check The figure shows the same angles measured in both degrees and radians Complete the missing measures y _ radians 90°; p °; 180°; 60°; radians _ radian °; p 3p radians °; 150°; 2p radians 30°; radians radians 210°; 0°; radians radians 225°; radian 330°; radians x radians 315°; radians radians °; 5p radians °; 4p radians 270°; 3p 66 Concept Check What is the exact radian measure of an angle measuring p degrees? Unless otherwise directed, give calculator approximations in answers in the rest of this exercise set Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r See Example 67 r = 12.3 cm, u = 69 r = 1.38 ft, u = 2p 5p radians radians 68 r = 0.892 cm, u = 70 r = 3.24 mi, u = 11p 10 7p radians radians 71 r = 4.82 m, u = 60° 72 r = 71.9 cm, u = 135° 73 r = 15.1 in., u = 210° 74 r = 12.4 ft, u = 330° Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line Assume that the radius of Earth is 6400 km See Example 75 Panama City, Panama, 9° N, and Pittsburgh, Pennsylvania, 40° N 76 Farmersville, California, 36° N, and Penticton, British Columbia, 49° N 77 New York City, New York, 41° N, and Lima, Peru, 12° S 78 Halifax, Nova Scotia, 45° N, and Buenos Aires, Argentina, 34° S 79 Latitude of Madison Madison, South Dakota, and Dallas, Texas, are 1200 km apart and lie on the same north-south line The latitude of Dallas is 33° N What is the latitude of Madison? 80 Latitude of Toronto Charleston, South Carolina, and Toronto, Canada, are 1100 km apart and lie on the same north-south line The latitude of Charleston is 33° N What is the latitude of Toronto? Work each problem See Examples and 81 Gear Movement Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure If the smaller gear rotates through an angle of 300°, through how many degrees does the larger gear rotate? M07_LIAL1953_06_GE_C06.indd 600 3.7 cm cm 31/08/16 4:54 pm www.downloadslide.com INDEX domain of, 234, 304 even, 292, 620, 680 exponential See Exponential function(s) exponential decay, 499, 501–502 exponential growth, 499–501 greatest integer, 279 identity, 275 increasing, 238 input-output machine illustration, 231 integrals of, 713 inverse See Inverse functions inverse cosecant, 728 inverse cosine, 726–727 inverse cotangent, 728 inverse secant, 728 inverse sine, 723–725 inverse tangent, 727–728 limit of, 43, 235 linear See Linear functions linear cost, 252 logarithmic See Logarithmic function(s) logistic, 510 notation, 235, 435 objective, 938 odd, 292, 619, 680 one-to-one, 432–434, 722 operations on, 304 periodic, 618 piecewise-defined, 278 polynomial See Polynomial function(s) profit, 252 quadratic See Quadratic function range of, 234 rational See Rational functions reciprocal, 387–388 revenue, 252 secant See Secant function sine See Sine function sinusoidal, 636 square root, 277 squaring, 276 step, 280 tangent See Tangent function tests for one-to-one, 434 trigonometric See Trigonometric functions vertical line test for, 233 Fundamental frequency, 745 Fundamental identities, 680–683 Fundamental principle of counting, 1080 Fundamental principle of fractions, 72 Fundamental rectangle of a hyperbola, 1009 Fundamental theorem of algebra, 356 Fundamental theorem of linear programming, 938 Future value, 117, 453, 1058–1059 ƒ1x2 notation, 235 G Galilei, Galileo, 152 Gauss, Carl Friedrich, 356 Gauss-Jordan method for solving linear systems, 891 Z04_LIAL1953_06_GE_IDX.indd 1187 General form of the equation of a circle, 222–223 Generalized principle of mathematical induction, 1075–1076 General term of a sequence, 1032 Geometric interpretation of circles, 986 Geometric progression, 1053 Geometric sequence common ratio of, 1053 definition of, 1053 infinite, 1058 nth term of, 1053 sum of infinitely many terms of, 1058 sum of terms of, 1055–1056 Geometric series definition of, 1055 infinite, 1056–1058 Geometry formulas, 1121 Geometry problems, 121, 150 Grade, 248, 405 Grade resistance, 556 Graph(s) of a circle, 222 of complex numbers, 819 compressed, 621 connecting with equations, 625, 648, 656 of cosecant function, 653 of cosine function, 620 of cotangent function, 644 of equations, 216 of exponential functions, 448–449 of a horizontal line, 246 horizontal translation of, 294, 366 of inequalities, 936 intercepts of, 216 of inverse cosecant function, 728 of inverse cosine function, 726–727 of inverse cotangent function, 728 of inverse functions, 439–440, 723 of inverse secant function, 728 of inverse sine function, 724–725 of inverse tangent function, 727–728 of linear inequalities in two variables, 934–937 of logarithmic functions, 463–467 of a parabola, 276, 987, 988 parametric, 850 of polar coordinates, 838 of polar equations, 840–844 of polynomial functions, 365–366 of quadratic functions, 330–332 of rational functions, 389–390 of the reciprocal function, 388 reflecting across a line, 288 reflecting across an axis, 288 of a relation, 230 of secant function, 652 shrinking of, 286–287 of sine function, 619 stretched, 621 stretching of, 286–287 summary of polar, 845 symmetry of, 289 of systems of inequalities, 936–937 of tangent function, 643 translations of, 293, 366 of a vertical line, 247 vertical translation of, 293, 366 1187 Graphing calculator method analyzing the path of a projectile, 854 for circles, 222 converting from trigonometric form to rectangular form, 820 decimal window on, 388 degree mode in, 535, 554 finding magnitude and direction angle of vectors, 809 finding the determinant of a matrix, 902 finding trigonometric function values with, 554 for fitting exponential curves to scatter diagrams, 456 graphing a cycloid, 852 graphing a plane curve defined parametrically, 850 graphing of polar equations, 841–844 performing row operations, 894 for rational functions, 389, 398 for the reciprocal function, 387 sine regression feature, 636–637 for solving linear equations in one variable, 267–268 for solving nonlinear systems, 923 solving problems involving angles of elevation, 571 for solving systems of equations using matrix inverses, 965 for solving systems of linear equations in two variables, 873, 875 solving trigonometric equations by linear methods, 738 square viewing window on, 222 standard viewing window on, 217 ZTrig viewing window, 621 Graphing techniques even and odd functions, 292 reflecting, 288 stretching and shrinking, 286–287 summary of, 297 symmetry, 289–291 translations, 297 Greatest common factor, 62 Greatest integer function, 279 Groundspeed, 803 Grouping, factoring by, 63 H Half-angle identities application of, 716–718 definition of, 715 general statement about, 716 simplifying expressions using, 717–718 trigonometric equations with, 742 verifying, 718 Half-life, 501 Half-plane boundary of, 934 definition of, 934 Harmonic motion, 660 Height of a projected object, 152, 158 Heron of Alexandria, 789 07/09/16 3:34 pm www.downloadslide.com 1188 INDEX Heron’s formula application of, 790 derivation of, 790–792 statement of, 789 Heron triangle, 797 Hipparchus, 563 Holding time, 90 Hooke’s Law, 415 Horizontal asymptote definition of, 387, 391 determining, 391 Horizontal axis, 819 Horizontal component of a vector, 809 Horizontal line equation of, 263, 266 graph of, 246 slope of, 249 Horizontal line test for one-to-one functions, 433 Horizontal parabola definition of, 986 graph of, 987 Horizontal shrinking of the graph of a function, 287 Horizontal stretching of the graph of a function, 287 Horizontal translation(s) of cosine function, 633 of cotangent function, 647–648 of a graph, 294 graphing, 631–633 of a graph of a polynomial function, 366 of sine function, 632 of tangent function, 647 Hubble constant, 271 Hyperbola(s) asymptotes of, 1008 center of, 1008 definition of, 721, 1007 eccentricity of, 1012, 1013 equations of, 1010 foci of, 1007 fundamental rectangle of, 1009 geometric definition of, 1007 geometric interpretation of, 986 translations of, 1011 transverse axis of, 1008 vertices of, 1008 Hyperbolic spiral, 857 Hypotenuse, 547 Hypotenuse of a right triangle, 151 Hypotrochoids, 852 I i definition of, 131 powers of, 136–137 i, j unit vectors, 812 Identities, trigonometric See Trigonometric identities Identity equations, 116, 117 Identity function, 275 Identity matrix, 960 Identity property for addition, 40 for multiplication, 40 Imaginary axis, 819 Z04_LIAL1953_06_GE_IDX.indd 1188 Imaginary numbers, 131 Imaginary part of a complex number, 131 Imaginary unit, 131 Impedance, 829 Impossible events, 1092 Inclination, angle of, 802 Inconsistent systems, 872, 894 Increasing function, 238 Independent equations, 872 Independent events, 1080 Independent variable, 229, 231 Index of a radical, 92, 94 of summation, 1036 Induction, mathematical, 1072–1077 Inequalities absolute value, 190 definition of, 42, 176 graphing, 934–937 linear, 176, 178, 934–937 nonstrict, 181, 935 properties of, 176 quadratic, 179 rational, 182 strict, 181, 935 three-part, 178 Infinite discontinuity, 397 Infinite geometric sequence definition of sum of terms, 1056–1058 formula for sum of terms of, 1058 Infinite geometric series, 1056–1058 Infinite sequence, 1032 Infinite series, 1035–1036 Infinite set, 28 Infinity, limits at, 396 Inflection point, 276 Initial point of a vector, 799 Initial side of an angle, 524 Inner product of vectors, 812 Input-output (function) machine, 231 Integer exponents, 50 Integers definition of, 35 relatively prime, 247 Integral, definite, 121, 335, 1037 Integrals of functions, 713 Intercepts of a graph, 216 Interest compound, 452 simple, 117 Intermediate value theorem, 373 Intersection of sets, 31, 32, 1092 Interval closed, 177 definition of, 177 disjoint, 177 notation, 177 open, 177 Inverse additive, 40 multiplicative, 40, 961 Inverse cosecant function definition of, 728 graph of, 728 Inverse cosine function definition of, 726 graph of, 726–727 Inverse cotangent function definition of, 728 graph of, 728 Inverse functions definition of, 435, 723 equation of, 436–437 facts about, 440 graphs of, 439–440, 723 notation for, 435, 723 review of concepts, 723 theorem on, 470 Inverse properties, 40 Inverse secant function definition of, 728 graph of, 728 Inverse sine function definition of, 724 graph of, 724–725 Inverse tangent function definition of, 727 graph of, 727–728 Inverse trigonometric equations, 751–754 Inverse trigonometric functions domains and ranges of, 729 equations with, 751–754 graphs of, 724–728 notation for, 723 summary of, 729 Inverse variation definition of, 411 as nth power, 411 Irrational numbers, 35 Is greater than, 42 Is less than, 42 J Joint variation, 411 Julia set, 822 Jump discontinuity, 397 K kth term of the binomial expansion, 1070 L Laffer curve, 406 Lambert’s law, 694 Latitude, 596 Law of cosines applications of, 787–788 applying to solve a triangle, 785 derivation and statement of, 785–786 Heron’s formula and, 790–792 Law of sines ambiguous case of, 774 applications of, 772–773 derivation and statement of, 771 Leading coefficient, 330 Least common denominator in fractions, 75 steps to find, 75 Legs of a right triangle, 151 Leibniz notation, 478 Lemniscates, 843, 845 Leonardo of Pisa, 1034 07/09/16 3:34 pm www.downloadslide.com INDEX Life table, 1101 Like radicals, 96 Like terms, 52 Limaçons, 844, 845 Limit(s) definition of, 43 of a function, 43, 235 at infinity, 396 notation, 43, 235 one-sided, 391 for the sum of the terms of an infinite geometric sequence, 1057 Line(s) See also Line segment(s) definition of, 524 equation of, 260 horizontal, 246, 249, 263, 266 parallel, 263 perpendicular, 263 polar form of, 840–841 secant, 307 slope of, 248 vertical, 247, 249, 263, 266 Linear cost function, 252 Linear equations definition of, 114, 872 in n unknowns, 872 standard form of, 247, 266 Linear equations in one variable definition of, 114 graphing calculator method for solving, 267–268 modeling with, 124–125 solution of, 114 solving by graphing, 267–268 Linear equations in two variables graphing calculator method for solving, 873, 875 modeling with, 266–267 point-slope form of, 260 slope-intercept form of, 261 standard form of, 247 summary of, 266 Linear functions definition of, 245 graphing, 245 point-slope form of, 260 slope-intercept form of, 261 x-intercept of graph of, 246 y-intercept of graph of, 246 Linear inequalities in one variable applications of, 178 solutions of, 176–177 standard form of, 176 Linear inequalities in two variables graphs of, 934–937 standard form of, 934 Linear methods for solving trigonometric equations, 738 Linear models, 124, 252 Linear programming constraints for, 938 definition of, 937 fundamental theorem of, 938 steps to solve problems with, 937 Linear regression, 267 Linear speed applications of, 611–613 definition of, 610 formula for, 610 Z04_LIAL1953_06_GE_IDX.indd 1189 Linear systems, 872 See also Systems of linear equations Line segment(s) definition of, 524 finding lengths of, 610 function values as lengths of, 609–610 Lissajous figure, 856 Literal equation, 117 Logarithmic differentiation, 468 Logarithmic equations definition of, 462 solving, 462–463, 490–492 steps to solve, 492 Logarithmic function(s) characteristics of graphs of, 465 graphs of, 463–467 modeling with, 476, 478, 493 standard form of, 463 Logarithms argument of, 461 base of, 461 change-of-base theorem for, 480 common, 475 definition of, 461 natural, 478 properties of, 467, 487 Logistic function, 510 LORAN, 1015 Lowest terms of a rational expression, 72 M Mach number, 721 Magnitude of a vector, 799 Major axis of an ellipse, 996 Mandelbrot set, 835 Mapping, 230 Marginal propensity to consume, 272 Mathematical induction generalized principle of, 1075–1076 principle of, 1072–1073 proof by, 1074 Mathematical model, 124, 252 Matrix (Matrices) addition of, 946 additive inverse of, 948 augmented, 890 cofactor of an element of, 903–904 column, 946 column of, 890 definition of, 890 determinant of, 902–903 diagonal form of, 891 dimensions of, 890 element of, 890 equal, 946 expansion by a given row or column, 904 identity, 960 minor of an element of, 903 modeling with, 953 multiplication of See Matrix multiplication multiplicative inverse of, 961 negative of, 948 1189 order of, 890 properties of, 946–954 reduced-row echelon form of, 891 row, 946 row of, 890 row transformations of, 890 scalar multiplication of, 949 sign array for, 905 singular, 965 size of, 890 square, 946 steps to find inverse of, 963 subtraction of, 948 zero, 947 Matrix inverses method for solving linear systems, 965 Matrix method for solving linear systems, 891 Matrix multiplication applications of, 953–954 definition of, 950 properties of, 949, 952 by scalars, 949 Maturity value, 117 Maximum point of a parabola, 334 Measure of an angle, 524, 554, 555 Members of a set, 28 Midpoint formula, 213 Minimum point of a parabola, 333 Minor axis of an ellipse, 996 Minor of a matrix element, 903 Minute, 526 Mixture problems, 122–123 Models absolute value, 192 cost-benefit, 81 exponential, 455 linear, 124, 252 linear systems, 879 logarithmic, 476, 478, 493 mathematical, 124, 252 matrix, 953 polynomial, 375–376 quadratic, 153, 336 rational, 398 Modulus of a complex number, 820 Monomial, 52 Motion problems, 121–122 Multiple-angle identity, deriving, 712 Multiple angles, trigonometric equations with, 743–744 Multiplication of binomials, 54–56 of complex numbers, 134 of functions, 304 identity property for, 40 of matrices See Matrix multiplication order of operations for, 37 of polynomials, 54–56 of radicals, 98 of rational expressions, 74–75 Multiplication property of equality, 114 Multiplication property of zero, 40 Multiplicative inverse definition of, 961 of a matrix, 961 of a real number, 40, 961 Multiplicity of zeros, 357 Mutually exclusive events, 1095 07/09/16 3:34 pm www.downloadslide.com 1190 INDEX N Napier, John, 469 Natural logarithms applications of, 478 definition of, 478 modeling with, 478 Natural numbers, 28, 35 Nautical mile, 603 Negation, order of operations for, 37 Negative of a matrix, 948 of a real number, 40 Negative angle, 524 Negative exponents, 81–82, 86 Negative reciprocals, 263 Negative slope, 251 Newton, 801 Newton’s law of cooling, 503 n-factorial (n!), 1066, 1081 Nonlinear systems of equations application of, 927–928 with complex solutions, 927 definition of, 923 elimination method for solving, 924 graphing calculator method for solving, 923 with nonreal solutions, 927 Nonstrict inequalities, 181, 935 Notation exponential, 37, 85 factorial, 1066, 1081 function, 235 interval, 177 inverse function, 435, 723 Leibniz, 478 limit, 43, 235 radical, 92 set-builder, 28 for sets of numbers, 28 sigma, 1035 summation, 1035 nth root of a complex number, 831 nth root theorem for complex numbers, 832 nth term of an arithmetic sequence, 1044 of a geometric sequence, 1053 of a sequence, 1032 Null set definition of, 29, 116 symbol for, 29, 116 Number line definition of, 35 distance between points on, 44 order on, 42 Number of zeros theorem, 356–357 Numbers complex See Complex numbers conjugate of, 99 counting, 28 decimals, 35 exact, 564 factors of, 36 imaginary, 131 integers, 35 irrational, 35 natural, 28, 35 negative reciprocals, 263 Z04_LIAL1953_06_GE_IDX.indd 1190 rational, 35 real, 35, 40 sets of, 28, 35 whole, 35 Numerator, rationalizing, 99 Numerical coefficient, 52 O Objective function, 938 Oblique asymptote definition of, 391, 393 determining, 391 Oblique triangle congruency and, 770–771 data required for solving, 770 definition of, 770 finding area of, 778–779 solving procedures for, 770–771 Obtuse angle, 524 Obtuse triangle, 771 Odd-degree polynomial functions, 367 Odd function, 292, 619, 680 Odds definition of, 1093 in favor of an event, 1093–1094 rules for, 1094 One-sided limits, 391 One-to-one functions definition of, 432, 722 horizontal line test for, 433 tests to determine, 434 Open interval, 177 Operations on complex numbers, 133–137 on functions, 304 order of, 37 with radicals, 96 on sets, 30–32 Opposite of a vector, 800 Opposite side to an angle, 547 Orbital period of a satellite, 848 Order descending, 53 of a matrix, 890 on a number line, 42 of operations, 37 Ordered pair, 210, 215 Ordered triple(s), 212, 878 Order symbols, 42 Ordinates, addition of, 656 Origin of rectangular coordinate system, 210 symmetry with respect to, 290 Orthogonal vectors, 814 Outcomes of an experiment, 1091 P Pair, ordered, 210, 215 Parabola(s) application of, 991 axis of, 331 definition of, 276, 330 directrix of, 988 eccentricity of, 1001 equations of, 986 focus of, 988 geometric definition of, 988 geometric interpretation of, 986 graphing by completing the square, 332 graph of, 276, 987 horizontal, 986, 987, 989 maximum point of, 334 minimum point of, 333 reflective property of, 991 translations of, 990 vertex formula for, 335 vertex of, 276, 331, 335, 986 vertical, 988 Parallel lines definition of, 263 slope of, 263 Parallelogram rule for vectors, 800 Parallelograms, properties of, 800 Parametric equations definition of, 631, 850 derivatives of, 566 of a plane curve, 850 Parentheses, 37 Partial fractions decomposition, 915 definition of, 915 Pascal, Blaise, 469, 1066 Pascal’s triangle, 1065–1066 Path of a projectile, 854 Pendulum, period of, 112 Perfect square trinomial, 65 Perfect triangle, 797 Period of cosecant function, 653 of cosine function, 620, 623 of cotangent function, 644 definition of, 623 of secant function, 652 of sine function, 619, 623 of tangent function, 643 Periodic function definition of, 618 period of, 618 Period life table, 1101 Period of a pendulum, 112 Permutations definition of, 1081 distinguishing from combinations, 1085–1086 formula for, 1081 Perpendicular lines definition of, 263 slope of, 263 pH, 476 Phase angle, 672 Phase shift, 631 Phelps, Phlash, 228 Photographic coordinates, 784 Pi (p), 103, 592 Piecewise-defined function, 278 Plane curve, 850 Plane trigonometry, 563 Plimpton 322 (tablet), 563 Point-slope form of the equation of a line, 260, 266 Poiseuille’s law, 416 Polar axis, 837 07/09/16 3:34 pm www.downloadslide.com INDEX Polar coordinates of a point definition of, 618, 837 graph of, 838 Polar coordinate system definition of, 837 polar axis of, 837 pole of, 837 Polar equations calculator graphing of, 841–844 classifying, 844–845 converting to rectangular equations, 844 definition of, 840 graphs of, 840–844 Polar form of a complex number, 820 of conic sections, 1113–1115 of lines and circles, 840–841 Polar grid, 841 Pole of a polar coordinate system, 837 Polynomial(s) addition of, 53 classifying expressions as, 53 definition of, 52 degree of, 52 descending order of, 53, 56 division algorithm for, 346 division of, 56–57 factored form of, 62 factoring of, 62 multiplication of, 54 prime, 62 special products of, 55 subtraction of, 53 synthetic division for, 347 term of, 52 in x, 52 zero, 330 Polynomial function(s) approximating real zeros of, 375 boundedness theorem for, 374 comprehensive graph of, 370 conjugate zeros theorem for, 358 definition of, 330 of degree n, 330 derivative of, 354 dominating term of, 330, 368 end behavior of graph of, 368, 369 of even degree, 367 factors of, 372 factor theorem for, 353 graph of, 365–366 intermediate value theorem for, 373 leading coefficient of, 330 modeling with, 375–376 number of zeros theorem for, 356–357 of odd degree, 367 rational zeros theorem for, 354 roots of, 350 solutions of, 350, 372 steps to graph, 370 turning points of graph of, 368 zero, 330 zeros of, 350, 359, 367, 372 Polynomial interpolation, 971 Polynomial models, 375–376 Population growth, 1034 Position vector, 809 Z04_LIAL1953_06_GE_IDX.indd 1191 Positive angle, 524 Positive slope, 251 Power property of equations, 165–166 Power rule for derivatives, 93 Power rules for exponents, 50, 86 Power series, 190 Powers of complex numbers, 830 Powers of i, 136–137 Present value, 453 Prime polynomial, 62 Principal root of a radical, 92 Principle of mathematical induction, 1072–1073 Probability of an event, 1091 basics of, 1091–1098 binomial, 1097 of compound events, 1095 definition of, 1091 properties of, 1097 of union of events, 1094–1095 Product of the sum and difference of two terms, 55 Product rule for exponents, 50, 86 Product theorem for complex numbers, 824 Product-to-sum identities, 713–714 Profit function, 252 Progression arithmetic, 1043 geometric, 1053 Proportional directly, 409 as nth power, 410–411 Proposed solutions, 162 Ptolemy, 563, 784 Pyramid frustum of, 60 volume of a frustum of, 60 Pythagorean identities, 541–542, 681 Pythagorean theorem, 151 Q Quadrantal angles, 527 Quadrants of a coordinate system, 210 Quadratic equations in one variable applications of, 150 completing the square to solve, 141 discriminant of, 145 factoring to solve, 139 modeling with, 153 quadratic formula for solving, 142–143 square root method for solving, 140 standard form of, 139 types of solutions for, 146 Quadratic formula discriminant of, 145 standard form of, 143 Quadratic function definition of, 330 graph of, 330–332 maximum of, 334 minimum of, 333 steps to graph, 335 Quadratic inequalities standard form of, 179 steps to solve, 179 1191 Quadratic models, 153, 336 Quadratic regression, 338 Queuing theory, 404 Quotient identities, 542–544, 681 Quotient rule for exponents, 82, 86 Quotient theorem for complex numbers, 825 R Radian, 592 Radian/degree relationship converting measures, 592–593 table of, 594 Radian measure applications of, 596–597 converting to degree measure, 592–593 definition of, 592 Radiative forcing, 479 Radical(s) addition of, 96 conjugate of, 99 index of, 92, 94 like, 96 multiplication of, 98 notation, 92 operations with, 96 principal root of, 92 rational exponent form of, 93 rationalizing the denominator of, 98 rationalizing the numerator of, 99 simplifying, 95–96 steps to simplify, 95 subtraction of, 96–97 summary of rules for, 94 symbol for, 92 unlike, 96 Radical equations power property and, 165–166 steps to solve, 166 Radical notation, 92 Radical symbol, 92 Radicand, 92 Radius of a circle, 221 Range(s) of a function, 234 of inverse circular functions, 729 of a relation, 231 of trigonometric functions, 540 Rate of change, average, 251 Rational equations, 162 Rational exponents definition of, 84 equations with, 168 radical form of, 92 Rational expressions addition of, 75 common factors of, 72, 73 decomposition of, 915 definition of, 72 division of, 74–75 domain of, 72 least common denominator of, 75 lowest terms of, 72 multiplication of, 74–75 subtraction of, 75–77 07/09/16 3:34 pm www.downloadslide.com 1192 INDEX Rational functions definition of, 386 features of, 390 graphing by calculator, 389, 398 graph of, 389–390 modeling with, 398 steps to graph, 393 Rational inequalities definition of, 182 steps to solve, 182 Rationalizing the denominator, 98 Rationalizing the numerator, 99 Rational numbers, 35 Rational zeros theorem, 354 Ray definition of, 524 endpoint of, 524 r cis u, 820 Real axis, 819 Real numbers additive inverse of, 40 definition of, 35 multiplicative inverse of, 40, 961 negative of, 40 properties of, 40 reciprocal of, 40 summary of properties of, 40 Real part of a complex number, 131 Reciprocal, 537 Reciprocal functions graphing by calculator, 387 graph of, 388 important features of, 388 Reciprocal identities, 537–538, 681 Reciprocal of a real number, 40 Rectangular coordinates, converting to polar coordinates, 838 Rectangular coordinate system, 210 Rectangular equations converting polar equations to, 844 definition of, 840 Rectangular form of a complex number converting to trigonometric form, 821 definition of, 819 Recursively defined sequence, 1033 Reduced-row echelon form of a matrix, 891 Reduction formula, 708 Reference angles definition of, 550 special angles as, 552–554 table of, 552 Reference arc, 604 Reflection of a graph across a line, 288 Reflection of a graph across an axis, 288 Reflective property of an ellipse, 1003 of a parabola, 991 Region of feasible solutions corner point of, 939 definition of, 938 vertex of, 939 Regression linear, 267 quadratic, 338 sine, 636–637 Related rates, 566 Z04_LIAL1953_06_GE_IDX.indd 1192 Relation definition of, 229 domain of, 231 graph of, 230 range of, 231 Relatively prime integers, 247 Remainder theorem, 349 Removable discontinuity, 397 Resistance, grade, 556 Resultant vectors, 800, 819 Revenue function, 252 Right angle, 524 Right triangle(s) hypotenuse of, 151, 547 legs of, 151 in Pythagorean theorem, 151 solving, 565 Right-triangle-based definitions of trigonometric functions, 547 Rise, 248 Roots of a complex number, 831–834 of an equation, 114 evaluating for radicals, 92 of an expression, 37 of polynomial functions, 350 Rose curve, 845 Rotation angles of, 1118 of axes, 1117–1119 definition of, 1117 Row matrix, 946 Row of a matrix, 890 Row transformations, 890 Rules for exponents, 50–52, 82, 86 Rules for odds, 1094 Run, 248 Rutherford, Ernest, 1015 S Sample space, 1091 Saturation constant, 1042 Scalar, 949 Scalar multiplication definition of, 949 properties of, 949 Scalars, 799 Scatter diagrams, 266, 456 Secant function characteristics of, 652 definition of, 533 domain of, 652 graph of, 652 inverse of, 728 period of, 652 range of, 540, 652 steps to graph, 654 Secant line, 307 Second, 526 Second-degree equation, 139 Sector of a circle area of, 597–598 definition of, 597 Segment of a line See Line segment(s) Semiperimeter, 789 Sequence(s) arithmetic, 1043, 1044, 1047 convergent, 1033 definition of, 1032 divergent, 1033 explicitly defined, 1033 Fibonacci, 1031, 1034 finite, 1032 general term of, 1032 geometric See Geometric sequence infinite, 1032 infinite geometric, 1056–1058 nth term of, 1032, 1044, 1053 recursively defined, 1033 terms of, 1032 Series arithmetic, 1047 definition of, 1035–1036 finite, 1035–1036 geometric, 1055 infinite, 1035–1036 infinite geometric, 1056–1058 power, 190 Set(s) complement of, 30, 32 definition of, 28 disjoint, 31 elements of, 28 empty, 29, 116 equal, 30 finite, 28 infinite, 28 intersection of, 31, 32, 1092 members of, 28 notation, 28 null, 29, 116 of numbers, 35 operations on, 30–32 subset of, 29 summary of operations on, 32 union of, 31, 32, 181, 1092 universal, 29 Set braces, 28 Set-builder notation, 28 Shrinking graphs graphing techniques for, 286–287 horizontally, 287 vertically, 287 Side adjacent to an angle, 547 Side-Angle-Angle (SAA) axiom, 770 Side-Angle-Side (SAS) axiom, 770 Side opposite an angle, 547 Side-Side-Side (SSS) axiom, 770 Sigma notation definition of, 1036 symbol for, 1036 Sign array for matrices, 905 Significant digits, 564–565 Signs of trigonometric function values, 538 Simple harmonic motion, 660 Simple interest formula, 117 Simplifying complex fractions, 77–78, 87–88 Simplifying radicals, 95–96 Sine function amplitude of, 621 characteristics of, 619 definition of, 533 difference identity for, 698–699 domain of, 619 double-angle identity for, 709–710 19/09/16 1:46 pm www.downloadslide.com INDEX graph of, 619 half-angle identity for, 715, 716 horizontal translation of, 632 inverse of, 724–725 period of, 619, 623 range of, 540, 619 steps to graph, 624, 635 sum identity for, 698–699 translating graphs of, 632, 635 Sine regression, 636–637 Sines, law of ambiguous case of, 774–777 applications of, 772–773 derivation of, 771 Sine wave, 619 Singular matrix, 965 Sinusoid, 619 Sinusoidal function, 636 Size of a matrix, 890 Slope(s) of a curve at a point, 248 definition of, 248 formula for, 248 of a horizontal line, 249 of a line, 248 negative, 251 of parallel lines, 263 of perpendicular lines, 263 point-slope form, 260, 266 positive, 251 slope-intercept form, 261, 266 undefined, 249 of a vertical line, 249 zero, 249, 251 Slope-intercept form of the equation of a line, 261, 266 Snell’s law, 765–766 Solar constant, 630 Solution(s) of an equation, 114 double, 141 of a polynomial function, 350, 372 proposed, 162 Solution set, 114 Solving applied problems, 120 Solving for a specified variable, 117, 145 Solving triangles, 565, 772–773, 775–777, 787–789 Sound waves, 630, 707 Special angles finding angle measures with, 554 reference angles as, 552–554 trigonometric function values of, 549–550 Special products of polynomials, 55 Speed angular, 610–613 definition of, 121 linear, 610–613 Spherical trigonometry, 563 Spiral, hyperbolic, 857 Spiral of Archimedes, 843–844 Square(s) of a binomial, 55 completing the, 141, 332 difference of, 66 Square matrix, 946 Square root function, 277 Z04_LIAL1953_06_GE_IDX.indd 1193 Square root method for solving quadratic equations, 140 Square root property, 140 Square viewing window on graphing calculators, 222 Squaring function, 276 Standard form of a complex number, 132, 819, 820 of the equation of a line, 260 of a linear equation, 247, 266 of a quadratic equation, 139 Standard position of an angle, 527 Standard viewing window on graphing calculators, 217 Statute mile, 603 Stefan-Boltzmann law, 416 Step function, 280 Stirling’s formula, 1072 Straight angle, 524 Stretching graphs graphing techniques for, 286–287 horizontal, 287 vertical, 287 Strict inequalities, 181, 935 Subscripts, 1036 Subset, 29 Substitution method definition of, 872 for factoring, 67–68 for solving linear systems, 873 Subtend an angle, 602 Subtense bar method, 570 Subtraction of complex numbers, 134 of functions, 304 of matrices, 948 order of operations for, 37 of polynomials, 53 of radicals, 96–97 of rational expressions, 75–77 of vectors, 800 Sum identity application of, 700–704 for cosine, 695–697 for sine, 698–699 for tangent, 699–700 Summation, index of, 1036 Summation notation definition of, 1036 with subscripts, 1036 Summation properties, 1037 Summation rules, 1037 Sum of cubes, 66 Sum of terms of an arithmetic sequence, 1047 of an infinite geometric sequence, 1058 of an infinite geometric series, 1056–1058 of a geometric sequence, 1055–1056 Sum-to-product identities, 714–715 Superelevation, 561 Supplementary angles, 525 Supply and demand, 889 Swokowski, Earl, 713 Symbols of order, 42 Symmetry example of, 209 graphing techniques for, 289 1193 of graphs, 289 with respect to origin, 290 with respect to x-axis, 289–290 with respect to y-axis, 289–290 tests for, 289–291 Synthetic division for polynomials, 347–349 Systems of equations definition of, 872 solutions of, 872 Systems of inequalities definition of, 936 graphing, 936–937 Systems of linear equations in three variables applications of, 881 elimination method for solving, 878 Gauss-Jordan method for solving, 893 geometric interpretation of, 877 matrix inverses method for solving, 965–966 modeling with, 879 solving, 877–878 Systems of linear equations in two variables applications of, 876 consistent, 872 Cramer’s rule for solving, 907–908 elimination method for solving, 873 equivalent, 873 Gauss-Jordan method for solving, 891 graphing calculator method for solving, 873, 875 inconsistent, 872 matrix method for solving, 891 solution set for, 872 substitution method for solving, 872 transformations of, 873 Systolic pressure, 628 T Tangent function characteristics of, 643 definition of, 533 difference identity for, 699–700 domain of, 643 double-angle identity for, 710 graph of, 643 half-angle identity for, 715, 716 horizontal translation of, 647 inverse of, 727–728 period of, 643 range of, 540, 643 steps to graph, 645 sum identity for, 699–700 vertical translation of, 647 Tangent line to a circle, 227 to a curve, 260 Tartaglia, Niccolo, 364 Term(s) coefficient of, 52 definition of, 52 dominating, 330, 368 like, 52 of a polynomial, 52 of a sequence, 1032 07/09/16 3:34 pm www.downloadslide.com 1194 INDEX Terminal point of a vector, 799 Terminal side of an angle, 524 Tests for symmetry, 289–291 Three-part inequalities, 178 Threshold sound, 477 Tolerance, 192 Traffic intensity, 398 Transformations of linear systems, 873 Transit, 574 Translation(s) combinations of, 635–636 definition of, 293 of an ellipse, 999–1000 of graphs, 293, 366 horizontal, 294, 631–633, 647 of a hyperbola, 1011 of a parabola, 990 summary of, 297 vertical, 293, 634, 647 Transverse axis of a hyperbola, 1008 Tree diagram, 1080 Trial, 1091 Triangle(s) acute, 771 area of, 778–779, 789 congruent, 770–771 Heron, 797 oblique, 770–771 obtuse, 771 Pascal’s, 1065–1066 perfect, 797 right See Right triangle(s) solving, 565–566, 772–773, 775–777, 787–789 Triangular form, 900 Triangulation method, 796 Trigonometric equations conditional, 738 with half-angles, 742 inverse, 751–754 linear methods for solving, 738 with multiple angles, 743–744 quadratic methods for solving, 739–740 solving using the quadratic formula, 740 solving by squaring, 741 solving by zero-factor property, 739 Trigonometric form of a complex number, 820 Trigonometric functions circular, 604 cofunctions of, 548, 697–698 combinations of translations of, 635–636 definitions of, 533 derivatives of, in calculus, 687 domains of, 606 inverses of, 722–729 ranges of, 540 right-triangle-based definitions of, 547–548 translations of, 631–637 unit circle and, 221 Trigonometric function values of acute angles, 547 finding with a calculator, 554–555 Z04_LIAL1953_06_GE_IDX.indd 1194 of nonquadrantal angles, 553 of quadrantal angles, 535–536 signs and ranges of, 538 of special angles, 549–550 undefined, 536 Trigonometric identities cofunction, 548, 697–698 difference, 695–697, 698–703, 700–703 double-angle, 709–712 even-odd, 680–681 fundamental, 680–683 half-angle, 715–718, 742 product-to-sum, 713–714 Pythagorean, 541–542, 681 quotient, 542–544, 681 reciprocal, 537–538, 681 solving conditional trigonometric equations using, 740–741 sum, 695–697, 698–703, 700–703 sum-to-product, 714–715 verifying, 687 Trigonometric models, 626, 636–637 Trigonometric substitution, 542 Trinomials definition of, 52 factoring, 64–65 perfect square, 65 Triples, ordered, 212, 878 Trochoid, 852 Turning points of polynomial function graphs, 368 U Undefined slope, 249 Union of sets, 31, 32, 181, 1092 Union of two events definition of, 1092 probability of, 1094–1095 Unit, imaginary, 131 Unit circle, 221, 604 Unit fractions, 612 Unit vector, 811 Universal set, 29 Unlike radicals, 96 Upper harmonics, 745, 750 V Value absolute, 42–43, 188 future, 117, 453, 1058–1059 maturity, 117 present, 453 Variable dependent, 229, 231 independent, 229, 231 solving for a specified, 117–118, 145 Variation combined, 411 constant of, 409 direct, 409 directly as the nth power, 410 inverse, 411 inversely as the nth power, 411 joint, 411 in sign, 360 steps to solve problems with, 410 Vector(s) algebraic interpretation of, 809–810 angle between, 814 applications of, 802–804 components of, 809 direction angle for, 809 dot product of, 812–814 equilibrant of, 801–802 horizontal component of, 809 i, j units, 812 initial point of, 799 inner product of, 812 magnitude of, 799 naming practices, 799 operations with, 811 opposite of, 800 orthogonal, 814 parallelogram rule for, 800 position, 809 quantities, 799 resultant of, 800, 819 symbol for, 799 terminal point of, 799 unit, 811 vertical component of, 809 zero, 800 Vector cross products, 903 Velocity constant, 121 in motion problems, 121 non-constant, 121 Venn diagram, 30, 1093 Verifying trigonometric identities, 687 Vertex (Vertices) of an angle, 524 of an ellipse, 996 of a hyperbola, 1008 of a parabola, 276, 331, 335, 986 of a polygonal region, 939 Vertex formula for a parabola, 335 Vertical asymptote behavior of graphs near, 396 definition of, 387, 391, 642 determining, 391 Vertical axis, 819 Vertical component of a vector, 809 Vertical line equation of, 263, 266 graph of, 247 slope of, 249 Vertical line test for functions, 233 Vertical parabola definition of, 988 graph of, 988 Vertical shrinking of the graph of a function, 287 Vertical stretching of the graph of a function, 287 Vertical translations of cosine function, 634 of cotangent function, 647–648 of a graph, 293, 366 graphing, 634 of tangent function, 647 07/09/16 3:34 pm www.downloadslide.com INDEX W Y Waiting-line theory, 404 Wave sine, 619 sound, 630, 707 Whispering gallery, 1003 Whole numbers, 35 Windchill formula, 102 Work rate problems, 164–165 y-axis definition of, 210 symmetry with respect to, 289–290 y-intercept, 216 X x-axis definition of, 210 symmetry with respect to, 289–290 x-intercept, 216, 372 xy-plane, 210 Z04_LIAL1953_06_GE_IDX.indd 1195 1195 Zero polynomial, 330 Zero slope, 249, 251 Zero vector, 800 ZTrig viewing window, 621 Z Zero(s) multiplication property of, 40 multiplicity of, 357 of polynomial functions, 350, 359, 367, 372 as whole number, 35 Zero exponent, 51 Zero-factor property, 72, 139, 739 Zero matrix, 947 07/09/16 3:34 pm www.downloadslide.com Geometry Formulas Square Rectangle Triangle Perimeter: P = 4s Perimeter: P = 2L + 2W Perimeter: P = a + b + c Area: Ꮽ = s Area: Ꮽ = LW Area: Ꮽ = bh s s s s a W L c h b Parallelogram Trapezoid Circle Perimeter: P = 2a + 2b Perimeter: P = a + b + c + B Diameter: d = 2r Area: Ꮽ = bh Area: Ꮽ = h1B + b2 Circumference: C = 2pr = pd Area: Ꮽ = pr b Chord b a h r c h a d B Cube Rectangular Solid Sphere pr Surface area: S = 4pr Volume: V = e 3 Volume: V = LWH Volume: V = Surface area: S = 6e Surface area: S = 2HW + 2LW + 2LH e r H e e Cone L W Right Circular Cylinder Right Pyramid 1 Volume: V = pr 2h Volume: V = pr 2h Volume: V = Bh 3 Surface area: S = pr 2r + h Surface area: S = 2prh + 2pr B = area of the base (excludes the base) (includes top and bottom) h Z05_LIAL1953_06_GE_BEP.indd 1196 h h r r 07/09/16 5:03 pm www.downloadslide.com Graphs of Functions 2.6 Identity Function 2.6 Squaring Function y 2.6 Cubing Function y y x –2 –1 x 2.6 Cube Root Function 2.6 Absolute Value Function 2.6 Greatest Integer Function y f (x) = [[x]] x –2 x y 1 –2 –3 –4 –1 x –4 –3 –2 x 3.5 Reciprocal Function y f (x) = | x | f (x) = √x –8 x y f (x) = √x 2 y f (x) = x3 f (x) = x f (x) = x 2.6 Square Root Function x –1 f (x) = 1x 3.4 Polynomial Functions y y y y 6 –3 –2 x x –4 –6 Degree 3; one real zero Degree 3; three real zeros –12 Z05_LIAL1953_06_GE_BEP.indd 1197 y f(x) = a x, a > (0, 1) x Degree 4; four real zeros –2 –1 –1 x Degree 6; three real zeros –2 4.3 Logarithmic Functions y y f(x) = log a x, < a < f(x) = log a x, a > (–1, 1a) (1, a) (–1, 1a) 4.2 Exponential Functions y –2 –1 f(x) = a x, < a < (0, 1) x (a, 1) (1, a) x (1, 0) , –1 a ( ) x (a, 1) (1, 0) x ( 1a , –1) 07/09/16 5:03 pm www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com Global edition College Algebra & Trigonometry For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition sixth edition Lial Hornsby Schneider Daniels College Algebra & Trigonometry Sixth edition Margaret L Lial • John Hornsby • David I Schneider • Callie J Daniels G LOBa l edition This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Pearson Global Edition Lial_06_1292151951_Final.indd 23/09/16 11:45 AM ... 360° 2P 21 0° 5P 330° 22 5° (– 22 , – 22 ) 4P3 24 0° 27 0° (– 12 , – √ 32 ) (√ 32 , 12 ) 300° (0, –1) 11P ( √ 32 , – 12 ) 5P ( 22 , – 22 ) ( 12 , – √ 32 ) 315° 3P x 7P The unit circle x2 + y2 = Figure... 21 20 25° 22 123 0° 23 135° 24 -740° 25 -800° 26 - 610° Convert each radian measure to degrees See Examples 2( a) and 2( b) 27 p 28 4p 29 5p 31 7p 32 15p 33 - 5p 34 - 7p 35 21 p 20 36 31p 20 ... www.downloadslide.com 605 6 .2 The Unit Circle and Circular Functions y (– 12 , √ 32 ) P (– 22 , 22 ) 2P3 120 ° ( – √3 , 2 ) ( – ) –1 90° 60° 5P 7P ( 12 , √ 32 ) P ( 22 , 22 ) P 3P 135° 45° P 30°