Algebra and trigonometry

EXPONENTS AND RADICALS x ᎏᎏn ϭ x mϪn x x Ϫn ϭ ᎏᎏn x x n xn aᎏyᎏb ϭ ᎏyᎏn x m x n ϭ x mϩn 1x m2 n ϭ x m n 1xy2 n ϭ x n y n n x1͞n ϭ ͙xෆ n n ෆ ϭ ͙xෆ ͙yෆ ͙xy n ͙xෆ Ίᎏᎏ๶y ϭ ᎏᎏ ͙ෆy x n n ͙͙ ෆෆ ෆ x ϭ͙͙ ෆෆ ෆ x ϭ ͙xෆ n m Formulas for area A, perimeter P, circumference C, volume V: Rectangle mn Box A ϭ l„ m x m͞n ϭ ͙xෆෆ ϭ Q͙xෆR n n m n GEOMETRIC FORMULAS m n V ϭ l„ h P ϭ 2l ϩ 2„ m h „ SPECIAL PRODUCTS 1x ϩ y2 ϭ x ϩ xy ϩ y Triangle 2 1x Ϫ y2 ϭ x Ϫ xy ϩ y 2 „ l l Pyramid V ϭ ᎏ13ᎏ A ϭ ᎏ12ᎏ bh 1x ϩ y23 ϭ x ϩ 3x y ϩ 3xy ϩ y 1x Ϫ y23 ϭ x Ϫ 3x y ϩ 3xy Ϫ y h h a FACTORING FORMULAS a b x Ϫ y ϭ 1x ϩ y21x Ϫ y2 Circle x ϩ 2xy ϩ y ϭ 1x ϩ y22 A ϭ ␲r x Ϫ 2xy ϩ y ϭ 1x Ϫ y22 x ϩ y ϭ 1x ϩ y21x Ϫ xy ϩ y 2 Sphere V ϭ ᎏ43ᎏ ␲ r C ϭ 2␲ r A ϭ 4␲ r x Ϫ y ϭ 1x Ϫ y21x ϩ xy ϩ y 2 r r QUADRATIC FORMULA Cylinder If ax ϩ bx ϩ c ϭ 0, then Ϫb Ϯ ͙ෆbෆ Ϫෆ ෆ 4ෆaෆc x ϭ ᎏᎏ 2a INEQUALITIES AND ABSOLUTE VALUE Cone V ϭ ␲ r 2h V ϭ ᎏ13ᎏ ␲ r 2h r h h If a Ͻ b and b Ͻ c, then a Ͻ c r If a Ͻ b, then a ϩ c Ͻ b ϩ c If a Ͻ b and c Ͼ 0, then ca Ͻ cb HERON’S FORMULA If a Ͻ b and c Ͻ 0, then ca Ͼ cb If a Ͼ 0, then ⏐x⏐ ϭ a means x ϭ a or x ϭ Ϫa ⏐x⏐ Ͻ a means Ϫa Ͻ x Ͻ a ⏐x⏐ Ͼ a means x Ͼ a or x Ͻ Ϫa B Area ϭ ͙ෆ s sෆෆ Ϫෆaෆ 21 sෆϪ ෆෆbෆ 21 sෆϪ ෆෆcෆ2 aϩbϩc where s ϭ ᎏᎏ c A a b C DISTANCE AND MIDPOINT FORMULAS GRAPHS OF FUNCTIONS Distance between P11x1 , y12 and P2 1x , y2 : Linear functions: f1x2 ϭ mx ϩ b ෆෆ d ϭ ͙ෆ 1xෆ ෆෆxෆ 1ෆ y2ෆ Ϫෆyෆ 2Ϫ 12 ෆϩ 12 ෆ Midpoint of P1P2: y x1 ϩ x2 y1 ϩ y2 a ᎏᎏ, ᎏᎏb 2 y b b LINES Slope of line through P11x1 , y12 and P2 1x , y2 Slope-intercept equation of line with slope m and y-intercept b y ϭ mx ϩ b Two-intercept equation of line with x-intercept a and y-intercept b x y ᎏᎏ ϩ ᎏᎏ ϭ a b Power functions: f1x2 ϭ x n y y x x Ï=≈ Ï=x£ f 1x2 ϭ ͙xළ n Root functions: LOGARITHMS y ϭ log a x Ï=mx+b Ï=b y Ϫ y1 ϭ m 1x Ϫ x12 Point-slope equation of line through P11x1, y12 with slope m x x y2 Ϫ y1 m ϭ ᎏᎏ x Ϫ x1 y y means a y ϭ x log a a x ϭ x a log a x ϭ x log a ϭ log a a ϭ log x ϭ log10 x ln x ϭ log e x log a xy ϭ log a x ϩ log a y log a aᎏxᎏb ϭ log a x Ϫ log a y y log a x b ϭ b log a x log b x ϭ x x Ï=£œx ∑ Ï=œ∑ x log a x log a b Reciprocal functions: f 1x2 ϭ 1/x n y y EXPONENTIAL AND LOGARITHMIC FUNCTIONS y y y=a˛ a>1 x x y=a˛ 00 b+ hyp opp ă _a x radians sin cos ␪ tan ␪ 0° 30° 45° 60° 90° 180° 270° ␲͞6 ␲͞4 ␲͞3 ␲͞2 ␲ 3␲͞2 1͞2 ͙ෆ2͞2 ͙ෆ3͞2 Ϫ1 ͙ෆ3͞2 ͙ෆ2͞2 1͞2 Ϫ1 0 ͙ෆ3͞3 ͙ෆ3 — — b _a One period b+ 2π x k One period period: 2␲͞k amplitude: ⏐a ⏐ ␪ a>0 a 2π k b adj SPECIAL VALUES OF THE TRIGONOMETRIC FUNCTIONS y ϭ a cos k 1x Ϫ b2 1k Ͼ 02 phase shift: b GRAPHS OF THE INVERSE TRIGONOMETRIC FUNCTIONS y ϭ sinϪ1 x _1 y ϭ cosϪ1 x y ϭ tanϪ1 x y y y π π π π _2 π x _1 x x π _2 FUNDAMENTAL IDENTITIES FORMULAS FOR REDUCING POWERS sec x ϭ cos x csc x ϭ sin x tan x ϭ sin x cos x cot x ϭ tan x Ϫ cos 2x sin x ϭ ᎏᎏ sin x ϩ cos x ϭ 1 ϩ tan x ϭ sec x ϩ cot2 x ϭ csc x sin1Ϫx2 ϭ Ϫsin x cos1Ϫx2 ϭ cos x tan1Ϫx2 ϭ Ϫtan x ␲ sin aᎏᎏ Ϫ xb ϭ cos x ␲ cos aᎏᎏ Ϫ xb ϭ sin x ␲ tan aᎏᎏ Ϫ xb ϭ cot x ␲ cot aᎏᎏ Ϫ xb ϭ tan x ␲ sec aᎏᎏ Ϫ xb ϭ csc x ␲ csc aᎏᎏ Ϫ xb ϭ sec x HALF-ANGLE FORMULAS ϩ cos u u cos ᎏᎏ ϭ Ϯ B 2 sin u Ϫ cos u u tan ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ sin u ϩ cos u PRODUCT-TO-SUM AND SUM-TO-PRODUCT IDENTITIES sin u cos √ ϭ ᎏ1ᎏ ͓sin1u ϩ √ ϩ sin1u Ϫ √ 2͔ cos u sin √ ϭ ᎏ1ᎏ ͓sin1u ϩ √ Ϫ sin1u Ϫ √ 2͔ REDUCTION IDENTITIES ␲ sin ax ϩ ᎏᎏb ϭ cos x sin1x ϩ ␲2 ϭ Ϫsin x cos1x ϩ ␲2 ϭ Ϫcos x ␲ cos ax ϩ ᎏᎏb ϭ Ϫsin x tan1x ϩ ␲2 ϭ tan x ␲ tan ax ϩ ᎏᎏb ϭ Ϫcot x ADDITION AND SUBTRACTION FORMULAS sin u sin √ ϭ ᎏᎏ ͓cos1u Ϫ √ Ϫ cos1u ϩ √ 2͔ xϩy xϪy sin x ϩ sin y ϭ sin ᎏᎏ cos ᎏᎏ 2 xϩy xϪy sin x Ϫ sin y ϭ cos ᎏᎏ sin ᎏᎏ 2 xϩy xϪy cos x Ϫ cos y ϭ Ϫ2 sin ᎏᎏ sin ᎏᎏ 2 sin1x Ϫ y2 ϭ sin x cos y Ϫ cos x sin y cos1x ϩ y2 ϭ cos x cos y Ϫ sin x sin y THE LAWS OF SINES AND COSINES cos1x Ϫ y2 ϭ cos x cos y ϩ sin x sin y tan x ϩ tan y tan1x ϩ y2 ϭ ᎏᎏ Ϫ tan x tan y cos u cos √ ϭ ᎏ1ᎏ ͓cos1u ϩ √ ϩ cos1u Ϫ √ 2͔ xϩy xϪy cos x ϩ cos y ϭ cos ᎏᎏ cos ᎏᎏ 2 sin1x ϩ y2 ϭ sin x cos y ϩ cos x sin y tan x Ϫ tan y tan1x Ϫ y2 ϭ ᎏᎏ ϩ tan x tan y DOUBLE-ANGLE FORMULAS The Law of Sines B a sin A sin B sin C ᎏᎏ ϭ ᎏᎏ ϭ ᎏᎏ a b c C cos 2x ϭ cos x Ϫ sin x 2 ϭ cos x Ϫ tan x tan 2x ϭ ᎏᎏ Ϫ tan x Ϫ cos x tan x ϭ ᎏᎏ ϩ cos x Ϫ cos u u sin ᎏᎏ ϭ Ϯ B 2 COFUNCTION IDENTITIES sin 2x ϭ sin x cos x ϩ cos 2x cos2 x ϭ ᎏᎏ ϭ Ϫ sin x c The Law of Cosines a ϭ b ϩ c Ϫ 2bc cos A b b ϭ a ϩ c Ϫ 2ac cos B c ϭ a ϩ b Ϫ 2ab cos C A ... analysis Stewart, Redlin, and Watson have also published Precalculus: Mathematics for Calculus, College Algebra, Trigonometry, and (with Phyllis Panman) College Algebra: Concepts and Contexts A BOUT... Test: Chapters 13 and 14 917 APPENDIX: Calculations and Significant Figures 919 ANSWERS A1 INDEX I1 This page intentionally left blank PREFACE For many students an Algebra and Trigonometry course... the text to graph and analyze functions, families of functions, and sequences; to calculate and graph regression curves; to perform matrix algebra; to graph linear inequalities; and other powerful