Right Triangles and Trigonometry


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1 9 Right Tringles nd Trigonometry 9.1 The Pythgoren Theorem 9. Specil Right Tringles 9.3 Similr Right Tringles 9.4 The Tngent Rtio 9.5 The Sine nd osine Rtios 9.6 Solving Right Tringles 9.7 Lw of Sines nd Lw of osines Lening Tower of Pis (p. 518) Skiing (p. 501) SEE the ig Ide Wshington Monument (p. 495) Rock Wll (p. 485) Fire Escpe (p. 473) Mthemticl Thinking: Mthemticlly proficient students cn pply the mthemtics they know to solve problems rising in everydy life, society, nd the workplce.
2 Mintining Mthemticl Proficiency Using Properties of Rdicls (.11.) Emple 1 Simplify = 64 Emple Simplify 4. 5 Fctor using the gretest perfect squre fctor. = 64 Product Property of Rdicls = 8 Simplify. 4 = = = Multiply by 5. 5 Product Property of Rdicls Simplify. Simplify the epression Solving Proportions (7.4.D) Emple 3 Solve 10 = = 3 = 10 3 = 30 = 30 = 15 Write the proportion. ross Products Property Multiply. Divide ech side by. Simplify. Solve the proportion = = = = = = STRT RESONING The Product Property of Rdicls llows you to simplify the squre root of product. re you ble to simplify the squre root of sum? of difference? Eplin. 465
3 Mthemticl Thinking ttending to Precision ore oncept Stndrd Position for Right Tringle In unit circle trigonometry, right tringle is in stndrd position when: 1. The hypotenuse is rdius of the circle of rdius 1 with center t the origin.. One leg of the right tringle lies on the is. 3. The other leg of the right tringle is perpendiculr to the is. Mthemticlly profi cient students disply, eplin, nd justify mthemticl ides nd rguments using precise mthemticl lnguge in written or orl communiction. (G.1.G) y Drwing n Isosceles Right Tringle in Stndrd Position Use dynmic geometry softwre to construct n isosceles right tringle in stndrd position. Wht re the ect coordintes of its vertices? Smple Points (0, 0) (0.71, 0.71) (0.71, 0) Segments = 1 = 0.71 = 0.71 ngle m = 45 To determine the ect coordintes of the vertices, lbel the length of ech leg. y the Pythgoren Theorem, which you will study in Section 9.1, + = 1. Solving this eqution yields = 1, or. So, the ect coordintes of the vertices re (0, 0), (, ), nd ( ), 0. Monitoring Progress 1. Use dynmic geometry softwre to construct right tringle with cute ngle mesures of 30 nd 60 in stndrd position. Wht re the ect coordintes of its vertices?. Use dynmic geometry softwre to construct right tringle with cute ngle mesures of 0 nd 70 in stndrd position. Wht re the pproimte coordintes of its vertices? 466 hpter 9 Right Tringles nd Trigonometry
4 9.1 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D G.9. The Pythgoren Theorem Essentil Question How cn you prove the Pythgoren Theorem? Proving the Pythgoren Theorem without Words Work with prtner.. Drw nd cut out right tringle with legs nd b, nd hypotenuse c. b. Mke three copies of your right tringle. rrnge ll four tringles to form lrge squre, s shown. c. Find the re of the lrge squre in terms of, b, nd c by summing the res of the tringles nd the smll squre. d. opy the lrge squre. Divide it into two smller squres nd two equllysized rectngles, s shown. b c c b b c b c b e. Find the re of the lrge squre in terms of nd b by summing the res of the rectngles nd the smller squres. b b f. ompre your nswers to prts (c) nd (e). Eplin how this proves the Pythgoren Theorem. b Proving the Pythgoren Theorem Work with prtner.. Drw right tringle with legs nd b, nd hypotenuse c, s shown. Drw the ltitude from to. Lbel the lengths, s shown. RESONING To be proficient in mth, you need to know nd fleibly use different properties of opertions nd objects. b h c d c D b. Eplin why, D, nd D re similr. d c. Write twocolumn proof using the similr tringles in prt (b) to prove tht + b = c. ommunicte Your nswer 3. How cn you prove the Pythgoren Theorem? 4. Use the Internet or some other resource to find wy to prove the Pythgoren Theorem tht is different from Eplortions 1 nd. Section 9.1 The Pythgoren Theorem 467
5 9.1 Lesson Wht You Will Lern ore Vocbulry Pythgoren triple, p. 468 Previous right tringle legs of right tringle hypotenuse Use the Pythgoren Theorem. Use the onverse of the Pythgoren Theorem. lssify tringles. Using the Pythgoren Theorem One of the most fmous theorems in mthemtics is the Pythgoren Theorem, nmed for the ncient Greek mthemticin Pythgors. This theorem describes the reltionship between the side lengths of right tringle. Theorem Theorem 9.1 Pythgoren Theorem In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. Proof Eplortions 1 nd, p. 467; E. 39, p. 488 c b c = + b STUDY TIP You my find it helpful to memorize the bsic Pythgoren triples, shown in bold, for stndrdized tests. Pythgoren triple is set of three positive integers, b, nd c tht stisfy the eqution c = + b. ore oncept ommon Pythgoren Triples nd Some of Their Multiples 3, 4, 5 6, 8, 10 9, 1, 15 3, 4, 5 5, 1, 13 10, 4, 6 15, 36, 39 5, 1, 13 8, 15, 17 16, 30, 34 4, 45, 51 8, 15, 17 7, 4, 5 14, 48, 50 1, 7, 75 7, 4, 5 The most common Pythgoren triples re in bold. The other triples re the result of multiplying ech integer in boldfced triple by the sme fctor. Using the Pythgoren Theorem Find the vlue of. Then tell whether the side lengths form Pythgoren triple. c = + b = = = 169 = 13 Pythgoren Theorem Substitute. Multiply. dd. Find the positive squre root. 5 1 The vlue of is 13. ecuse the side lengths 5, 1, nd 13 re integers tht stisfy the eqution c = + b, they form Pythgoren triple. 468 hpter 9 Right Tringles nd Trigonometry
6 Using the Pythgoren Theorem Find the vlue of. Then tell whether the side lengths form Pythgoren triple. 7 c = + b Pythgoren Theorem 14 = 7 + Substitute. 196 = 49 + Multiply. 147 = Subtrct 49 from ech side. 147 = Find the positive squre root = Product Property of Rdicls 7 3 = Simplify. 14 The vlue of is 7 3. ecuse 7 3 is not n integer, the side lengths do not form Pythgoren triple. Solving RelLife Problem The skyscrpers shown re connected by skywlk with support bems. Use the Pythgoren Theorem to pproimte the length of ech support bem. Ech support bem forms the hypotenuse of right tringle. The right tringles re congruent, so the support bems re the sme length. = (3.6) + (47.57) Pythgoren Theorem = (3.6) + (47.57) Find the positive squre root m Use clcultor to pproimte. The length of ech support bem is bout 5.95 meters. 3.6 m support bems m Monitoring Progress Help in English nd Spnish t igidesmth.com Find the vlue of. Then tell whether the side lengths form Pythgoren triple n nemometer is device used to mesure wind speed. The nemometer shown is ttched to the top of pole. Support wires re ttched to the pole 5 feet bove the ground. Ech support wire is 6 feet long. How fr from the bse of the pole is ech wire ttched to the ground? 6 ft d 5 ft Section 9.1 The Pythgoren Theorem 469
7 Using the onverse of the Pythgoren Theorem The converse of the Pythgoren Theorem is lso true. You cn use it to determine whether tringle with given side lengths is right tringle. Theorem Theorem 9. onverse of the Pythgoren Theorem If the squre of the length of the longest side of tringle is equl to the sum of the squres of the lengths of the other two sides, then the tringle is right tringle. If c = + b, then is right tringle. Proof E. 39, p. 474 c b Verifying Right Tringles Tell whether ech tringle is right tringle b SELETING TOOLS Use clcultor to determine tht is the length of the longest side in prt (). 113 Let c represent the length of the longest side of the tringle. heck to see whether the side lengths stisfy the eqution c = + b.. ( 113 ) =? =? = 113 The tringle is right tringle. b. ( 4 95 ) =? ( 95 ) =? =? The tringle is not right tringle. Monitoring Progress Help in English nd Spnish t igidesmth.com Tell whether the tringle is right tringle hpter 9 Right Tringles nd Trigonometry
8 lssifying Tringles The onverse of the Pythgoren Theorem is used to determine whether tringle is right tringle. You cn use the theorem below to determine whether tringle is cute or obtuse. Theorem Theorem 9.3 Pythgoren Inequlities Theorem For ny, where c is the length of the longest side, the following sttements re true. If c < + b, then is cute. If c > + b, then is obtuse. b c b c REMEMER The Tringle Inequlity Theorem (Theorem 6.11) on pge 343 sttes tht the sum of the lengths of ny two sides of tringle is greter thn the length of the third side. lssifying Tringles Verify tht segments with lengths of 4.3 feet, 5. feet, nd 6.1 feet form tringle. Is the tringle cute, right, or obtuse? c < + b Proof Es. 4 nd 43, p. 474 Step 1 Use the Tringle Inequlity Theorem (Theorem 6.11) to verify tht the segments form tringle. c > + b >? >? >? > > > 4.3 The segments with lengths of 4.3 feet, 5. feet, nd 6.1 feet form tringle. Step lssify the tringle by compring the squre of the length of the longest side with the sum of the squres of the lengths of the other two sides. c + b ompre c with + b Substitute Simplify < c is less thn + b. The segments with lengths of 4.3 feet, 5. feet, nd 6.1 feet form n cute tringle. Monitoring Progress Help in English nd Spnish t igidesmth.com 6. Verify tht segments with lengths of 3, 4, nd 6 form tringle. Is the tringle cute, right, or obtuse? 7. Verify tht segments with lengths of.1,.8, nd 3.5 form tringle. Is the tringle cute, right, or obtuse? Section 9.1 The Pythgoren Theorem 471
9 9.1 Eercises Tutoril Help in English nd Spnish t igidesmth.com Vocbulry nd ore oncept heck 1. VOULRY Wht is Pythgoren triple?. DIFFERENT WORDS, SME QUESTION Which is different? Find both nswers. Find the length of the longest side. Find the length of the hypotenuse. 3 Find the length of the longest leg. 4 Find the length of the side opposite the right ngle. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, find the vlue of. Then tell whether the side lengths form Pythgoren triple. (See Emple 1.) In Eercises 7 10, find the vlue of. Then tell whether the side lengths form Pythgoren triple. (See Emple.) ERROR NLYSIS In Eercises 11 nd 1, describe nd correct the error in using the Pythgoren Theorem (Theorem 9.1) c = + b = = (7 + 4) = 31 = 31 c = + b = = = 776 = hpter 9 Right Tringles nd Trigonometry
10 13. MODELING WITH MTHEMTIS The fire escpe forms right tringle, s shown. Use the Pythgoren Theorem (Theorem 9.1) to pproimte the distnce between the two pltforms. (See Emple 3.) In Eercises 1 8, verify tht the segment lengths form tringle. Is the tringle cute, right, or obtuse? (See Emple 5.) 1. 10, 11, nd 14. 6, 8, nd ft 3. 1, 16, nd , 0, nd , 6.7, nd , 8., nd , 30, nd , 15, nd ft 14. MODELING WITH MTHEMTIS The bckbord of the bsketbll hoop forms right tringle with the supporting rods, s shown. Use the Pythgoren Theorem (Theorem 9.1) to pproimte the distnce between the rods where they meet the bckbord. 9. MODELING WITH MTHEMTIS In bsebll, the lengths of the pths between consecutive bses re 90 feet, nd the pths form right ngles. The plyer on first bse tries to stel second bse. How fr does the bll need to trvel from home plte to second bse to get the plyer out? 30. RESONING You re mking cnvs frme for pinting using stretcher brs. The rectngulr pinting will be 10 inches long nd 8 inches wide. Using ruler, how cn you be certin tht the corners of the frme re 90? 13.4 in. 9.8 in. In Eercises 15 0, tell whether the tringle is right tringle. (See Emple 4.) In Eercises 31 34, find the re of the isosceles tringle m 10 cm h 16 m h 17 m 10 cm ft h 3 ft 0 ft cm 50 m h 50 m m Section 9.1 The Pythgoren Theorem 473
11 35. NLYZING RELTIONSHIPS Justify the Distnce Formul using the Pythgoren Theorem (Thm. 9.1). 36. HOW DO YOU SEE IT? How do you know is right ngle without using the Pythgoren Theorem (Theorem 9.1)? PROLEM SOLVING You re mking kite nd need to figure out how much binding to buy. You need the binding for the perimeter of the kite. 1 in. The binding comes in pckges of two yrds. How mny pckges should you buy? 38. PROVING THEOREM Use the Pythgoren Theorem (Theorem 9.1) to prove the HypotenuseLeg (HL) ongruence Theorem (Theorem 5.9). 39. PROVING THEOREM Prove the onverse of the Pythgoren Theorem (Theorem 9.). (Hint: Drw with side lengths, b, nd c, where c is the length of the longest side. Then drw right tringle with side lengths, b, nd, where is the length of the hypotenuse. ompre lengths c nd.) 40. THOUGHT PROVOKING onsider two integers m nd n, where m > n. Do the following epressions produce Pythgoren triple? If yes, prove your nswer. If no, give counteremple. mn, m n, m + n 8 15 in. 1 in. 0 in. 41. MKING N RGUMENT Your friend clims 7 nd 75 cnnot be prt of Pythgoren triple becuse does not equl positive integer squred. Is your friend correct? Eplin your resoning. 4. PROVING THEOREM opy nd complete the proof of the Pythgoren Inequlities Theorem (Theorem 9.3) when c < + b. Given In, c < + b, where c is the length of the longest side. PQR hs side lengths, b, nd, where is the length of the hypotenuse, nd R is right ngle. Prove is n cute tringle. STTEMENTS c b Q R 1. In, c < + b, where c is the length of the longest side. PQR hs side lengths, b, nd, where is the length of the hypotenuse, nd R is right ngle. P b RESONS b =. 3. c < c < 4. Tke the positive squre root of ech side. 5. m R = m < m R 6. onverse of the Hinge Theorem (Theorem 6.13) 7. m < is n cute ngle is n cute tringle PROVING THEOREM Prove the Pythgoren Inequlities Theorem (Theorem 9.3) when c > + b. (Hint: Look bck t Eercise 4.) Mintining Mthemticl Proficiency Simplify the epression by rtionlizing the denomintor. (Skills Review Hndbook) Reviewing wht you lerned in previous grdes nd lessons hpter 9 Right Tringles nd Trigonometry
12 9. TEXS ESSENTIL KNOWLEDGE ND SKILLS G.9. USING PREISE MTHEMTIL LNGUGE To be proficient in mth, you need to epress numericl nswers with degree of precision pproprite for the problem contet. Specil Right Tringles Essentil Question Wht is the reltionship mong the side lengths of tringles? tringles? Side Rtios of n Isosceles Right Tringle Work with prtner.. Use dynmic geometry softwre to construct n isosceles right tringle with leg length of 4 units. b. Find the cute ngle mesures. Eplin why this tringle is clled tringle. c. Find the ect rtios of the side lengths (using squre roots). = = = d. Repet prts () nd (c) for severl other isosceles right tringles. Use your results to write conjecture bout the rtios of the side lengths of n isosceles right tringle. Smple Points (0, 4) (4, 0) (0, 0) Segments = 5.66 = 4 = 4 ngles m = 45 m = 45 Side Rtios of Tringle Work with prtner.. Use dynmic geometry softwre to construct right tringle with cute ngle mesures of 30 nd 60 ( tringle), where the shorter leg length is 3 units. b. Find the ect rtios Smple 5 of the side lengths (using squre roots). = = = c. Repet prts () nd (b) for severl other tringles. Use your results to write conjecture bout the rtios of the side lengths of tringle. ommunicte Your nswer 3. Wht is the reltionship mong the side lengths of tringles? tringles? Points (0, 5.0) (3, 0) (0, 0) Segments = 6 = 3 = 5.0 ngles m = 30 m = 60 Section 9. Specil Right Tringles 475
13 9. Lesson Wht You Will Lern ore Vocbulry Previous isosceles tringle Find side lengths in specil right tringles. Solve rellife problems involving specil right tringles. Finding Side Lengths in Specil Right Tringles tringle is n isosceles right tringle tht cn be formed by cutting squre in hlf digonlly. REMEMER rdicl with inde is in simplest form when no rdicnds hve perfect squres s fctors other thn 1, no rdicnds contin frctions, nd no rdicls pper in the denomintor of frction. Theorem Theorem Tringle Theorem In tringle, the hypotenuse is times s long s ech leg. Finding Side Lengths in Tringles Find the vlue of. Write your nswer in simplest form.. b Proof E. 19, p. 480 hypotenuse = leg. y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must be 45, so the tringle is tringle. hypotenuse = leg Tringle Theorem = 8 = 8 The vlue of is 8. Substitute. Simplify. b. y the se ngles Theorem (Theorem 5.6) nd the orollry to the Tringle Sum Theorem (orollry 5.1), the tringle is tringle. hypotenuse = leg Tringle Theorem 5 = Substitute. 5 = Divide ech side by. 5 = Simplify. The vlue of is hpter 9 Right Tringles nd Trigonometry
14 Theorem Theorem Tringle Theorem In tringle, the hypotenuse is twice s long s the shorter leg, nd the longer leg is 3 times s long s the shorter leg. Proof E. 1, p hypotenuse = shorter leg longer leg = shorter leg 3 REMEMER ecuse the ngle opposite 9 is lrger thn the ngle opposite, the leg with length 9 is longer thn the leg with length by the Tringle Lrger ngle Theorem (Theorem 6.10). Finding Side Lengths in Tringle Find the vlues of nd y. Write your nswer in simplest form. Step 1 Find the vlue of. longer leg = shorter leg = 3 Substitute Tringle Theorem 9 = Divide ech side by = Multiply by = Multiply frctions = Simplify. y The vlue of is 3 3. Step Find the vlue of y. hypotenuse = shorter leg y = 3 3 y = Tringle Theorem Substitute. Simplify. The vlue of y is 6 3. Monitoring Progress Help in English nd Spnish t igidesmth.com Find the vlue of the vrible. Write your nswer in simplest form. 1.. y h 4 Section 9. Specil Right Tringles 477
15 Solving RelLife Problems Modeling with Mthemtics The rod sign is shped like n equilterl tringle. Estimte the re of the sign by finding the re of the equilterl tringle. First find the height h of the tringle by dividing it into two tringles. The length of the longer leg of one of these tringles is h. The length of the shorter leg is 18 inches. h = 18 3 = Tringle Theorem Use h = 18 3 to find the re of the equilterl tringle. re = 1 bh = 1 (36) ( 18 3 ) The re of the sign is bout 561 squre inches. 36 in. YIELD 18 in. 18 in h 36 in. 36 in. Finding the Height of Rmp tipping pltform is rmp used to unlod trucks. How high is the end of n 80foot rmp when the tipping ngle is 30? 45? rmp 80 ft height of rmp tipping ngle 14 ft 60 When the tipping ngle is 30, the height h of the rmp is the length of the shorter leg of tringle. The length of the hypotenuse is 80 feet. 80 = h Tringle Theorem 40 = h Divide ech side by. When the tipping ngle is 45, the height h of the rmp is the length of leg of tringle. The length of the hypotenuse is 80 feet. 80 = h Tringle Theorem 80 = h Divide ech side by h Use clcultor. When the tipping ngle is 30, the rmp height is 40 feet. When the tipping ngle is 45, the rmp height is bout 56 feet 7 inches. Monitoring Progress Help in English nd Spnish t igidesmth.com 5. The logo on recycling bin resembles n equilterl tringle with side lengths of 6 centimeters. pproimte the re of the logo. 6. The body of dump truck is rised to empty lod of snd. How high is the 14footlong body from the frme when it is tipped upwrd by 60 ngle? 478 hpter 9 Right Tringles nd Trigonometry
16 9. Eercises Tutoril Help in English nd Spnish t igidesmth.com Vocbulry nd ore oncept heck 1. VOULRY Nme two specil right tringles by their ngle mesures.. WRITING Eplin why the cute ngles in n isosceles right tringle lwys mesure 45. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, find the vlue of. Write your nswer in simplest form. (See Emple 1.) In Eercises 7 10, find the vlues of nd y. Write your nswers in simplest form. (See Emple.) y y ERROR NLYSIS In Eercises 11 nd 1, describe nd correct the error in finding the length of the hypotenuse y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must be 60. So, the tringle is tringle. hypotenuse = shorter leg 3 = 7 3 y y y the Tringle Sum Theorem (Theorem 5.1), the mesure of the third ngle must be 45. So, the tringle is tringle. hypotenuse = leg leg = 5 So, the length of the hypotenuse is 5 units. In Eercises 13 nd 14, sketch the figure tht is described. Find the indicted length. Round deciml nswers to the nerest tenth. 13. The side length of n equilterl tringle is 5 centimeters. Find the length of n ltitude. 14. The perimeter of squre is 36 inches. Find the length of digonl. In Eercises 15 nd 16, find the re of the figure. Round deciml nswers to the nerest tenth. (See Emple 3.) ft m 60 5 m 17. PROLEM SOLVING Ech hlf of the drwbridge is bout 84 feet long. How high does the drwbridge rise when is 30? 45? 60? (See Emple 4.) 84 ft 5 m 4 m So, the length of the hypotenuse is 7 3 units. Section 9. Specil Right Tringles 479
17 18. MODELING WITH MTHEMTIS nut is shped like regulr hegon with side lengths of 1 centimeter. Find the vlue of. (Hint: regulr hegon cn be divided into si congruent tringles.) 1 cm. THOUGHT PROVOKING specil right tringle is right tringle tht hs rtionl ngle mesures nd ech side length contins t most one squre root. There re only three specil right tringles. The digrm below is clled the illes rectngle. Lbel the sides nd ngles in the digrm. Describe ll three specil right tringles. 19. PROVING THEOREM Write prgrph proof of the Tringle Theorem (Theorem 9.4). Given DEF is D tringle. Prove The hypotenuse is 45 times s long 45 s ech leg. F E 0. HOW DO YOU SEE IT? The digrm shows prt of the Wheel of Theodorus WRITING Describe two wys to show tht ll isosceles right tringles re similr to ech other. 4. MKING N RGUMENT Ech tringle in the digrm is tringle. t Stge 0, the legs of the tringle re ech 1 unit long. Your brother clims the lengths of the legs of the tringles dded re hlved t ech stge. So, the length of leg of tringle dded in Stge 8 will be 1 unit. Is your 56 brother correct? Eplin your resoning Which tringles, if ny, re tringles? b. Which tringles, if ny, re tringles? Stge 0 Stge 1 Stge 1. PROVING THEOREM Write prgrph proof of the Tringle Theorem (Theorem 9.5). (Hint: onstruct JML congruent to JKL.) Given JKL is tringle. Prove The hypotenuse is twice s long s the shorter leg, nd the longer leg is 3 times s long s the shorter leg. Mintining Mthemticl Proficiency Find the vlue of. (Section 8.1) J 6. DEF LMN 7. QRS K L M Stge 3 Stge 4 5. USING STRUTURE TUV is tringle, where two vertices re U(3, 1) nd V( 3, 1), UV is the hypotenuse, nd point T is in Qudrnt I. Find the coordintes of T. Reviewing wht you lerned in previous grdes nd lessons E 1 D N 0 F 30 L M S 4 R Q 480 hpter 9 Right Tringles nd Trigonometry
18 9.3 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.8. G.8. Similr Right Tringles Essentil Question How re ltitudes nd geometric mens of right tringles relted? Writing onjecture Work with prtner.. Use dynmic geometry softwre to construct right, s shown. Drw D so tht it is n ltitude from the right ngle to the hypotenuse of D Points (0, 5) (8, 0) (0, 0) D(.5, 3.6) Segments = 9.43 = 8 = 5 MKING MTHEMTIL RGUMENTS To be proficient in mth, you need to understnd nd use stted ssumptions, definitions, nd previously estblished results in constructing rguments. b. The geometric men of two positive numbers nd b is the positive number tht stisfies =. is the geometric men of nd b. b Write proportion involving the side lengths of D nd D so tht D is the geometric men of two of the other side lengths. Use similr tringles to justify your steps. c. Use the proportion you wrote in prt (b) to find D. d. Generlize the proportion you wrote in prt (b). Then write conjecture bout how the geometric men is relted to the ltitude from the right ngle to the hypotenuse of right tringle. Work with prtner. Use spredsheet to find the rithmetic men nd the geometric men of severl pirs of positive numbers. ompre the two mens. Wht do you notice? ommunicte Your nswer ompring Geometric nd rithmetic Mens D b rithmetic Men Geometric Men How re ltitudes nd geometric mens of right tringles relted? Section 9.3 Similr Right Tringles 481
19 9.3 Lesson Wht You Will Lern ore Vocbulry geometric men, p. 484 Previous ltitude of tringle similr figures Identify similr tringles. Solve rellife problems involving similr tringles. Use geometric mens. Identifying Similr Tringles When the ltitude is drwn to the hypotenuse of right tringle, the two smller tringles re similr to the originl tringle nd to ech other. Theorem Theorem 9.6 Right Tringle Similrity Theorem If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to ech other. D D, D, nd D D. Proof E. 45, p. 488 D D Identifying Similr Tringles Identify the similr tringles in the digrm. U S R T Sketch the three similr right tringles so tht the corresponding ngles nd sides hve the sme orienttion. T S S T U R U R T TSU RTU RST Monitoring Progress Help in English nd Spnish t igidesmth.com Identify the similr tringles. 1. Q. E H F T S R G 48 hpter 9 Right Tringles nd Trigonometry
20 Solving RelLife Problems Modeling with Mthemtics roof hs cross section tht is right tringle. The digrm shows the pproimte dimensions of this cross section. Find the height h of the roof. Y 5.5 m h 3.1 m Z 6.3 m W X 1. Understnd the Problem You re given the side lengths of right tringle. You need to find the height of the roof, which is the ltitude drwn to the hypotenuse.. Mke Pln Identify ny similr tringles. Then use the similr tringles to write proportion involving the height nd solve for h. 3. Solve the Problem Identify the similr tringles nd sketch them. Z Z OMMON ERROR Notice tht if you tried to write proportion using XYW nd YZW, then there would be two unknowns, so you would not be ble to solve for h. 3.1 m X Y h W 5.5 m Y h W X 6.3 m XYW YZW XZY ecuse XYW XZY, you cn write proportion. 3.1 m Y 5.5 m YW ZY = XY XZ orresponding side lengths of similr tringles re proportionl. h 5.5 = Substitute. h.7 Multiply ech side by 5.5. The height of the roof is bout.7 meters. 4. Look ck ecuse the height of the roof is leg of right YZW nd right XYW, it should be shorter thn ech of their hypotenuses. The lengths of the two hypotenuses re YZ = 5.5 nd XY = 3.1. ecuse.7 < 3.1, the nswer seems resonble. Monitoring Progress Find the vlue of. Help in English nd Spnish t igidesmth.com 3. E 3 G H 4 5 F 4. J 13 1 K L 5 M Section 9.3 Similr Right Tringles 483
21 Using Geometric Men ore oncept Geometric Men The geometric men of two positive numbers nd b is the positive number tht stisfies = b. So, = b nd = b. Finding Geometric Men Find the geometric men of 4 nd 48. = b Definition of geometric men = 4 48 Substitute 4 for nd 48 for b. = 4 48 Tke the positive squre root of ech side. = 4 4 Fctor. = 4 Simplify. The geometric men of 4 nd 48 is D In right, ltitude D is drwn to the hypotenuse, forming two smller right tringles tht re similr to. From the Right Tringle Similrity Theorem, you know tht D D. ecuse the tringles re similr, you cn write nd simplify the following proportions involving geometric mens. D D = D D D = D = D = D D = D = D D D Theorems Theorem 9.7 Geometric Men (ltitude) Theorem In right tringle, the ltitude from the right ngle to the hypotenuse divides the hypotenuse into two segments. The length of the ltitude is the geometric men of the lengths of the two segments of the hypotenuse. D Proof E. 41, p. 488 D = D D Theorem 9.8 Geometric Men (Leg) Theorem In right tringle, the ltitude from the right ngle to the hypotenuse divides the hypotenuse into two segments. The length of ech leg of the right tringle is the geometric men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djcent to the leg. D = D = D Proof E. 4, p hpter 9 Right Tringles nd Trigonometry
22 Using Geometric Men OMMON ERROR In Emple 4(b), the Geometric Men (Leg) Theorem gives y = (5 + ), not y = 5 (5 + ), becuse the side with length y is djcent to the segment with length. Find the vlue of ech vrible b. y 5. pply the Geometric Men b. pply the Geometric Men (ltitude) Theorem. (Leg) Theorem. = 6 3 y = (5 + ) = 18 y = 7 = 18 y = 14 = 9 y = 14 = 3 The vlue of y is 14. The vlue of is 3. Using Indirect Mesurement To find the cost of instlling rock wll in your school gymnsium, you need to find the height of the gym wll. You use crdbord squre to line up the top nd bottom om of the gym wll. Your friend mesures the verticl distnce from the ground to your eye nd the horizontl distnce from you to the gym wll. pproimte the height of the gym wll. y the Geometric Men (ltitude) Theorem, you know tht 8.5 is the geometric men of w nd = w 5 Geometric Men (ltitude) Theorem 7.5 = 5w Squre = w Divide ech side by 5. The height of the wll is 5 + w = = feet. 8.5 ft w ft 5 ft 4 9 Monitoring Progress Find the geometric men of the two numbers. Help in English nd Spnish t igidesmth.com 5. 1 nd nd nd Find the vlue of in the tringle t the left. 9. WHT IF? In Emple 5, the verticl distnce from the ground to your eye is 5.5 feet nd the distnce from you to the gym wll is 9 feet. pproimte the height of the gym wll. Section 9.3 Similr Right Tringles 485
23 9.3 Eercises Tutoril Help in English nd Spnish t igidesmth.com Vocbulry nd ore oncept heck 1. OMPLETE THE SENTENE If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd.. WRITING In your own words, eplin geometric men. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 nd 4, identify the similr tringles. (See Emple 1.) 3. F E In Eercises 11 18, find the geometric men of the two numbers. (See Emple 3.) nd nd 16 H G nd nd nd nd 8 4. M nd nd 45 L N K In Eercises 5 10, find the vlue of. (See Emple.) Q 5 W Y T X 4 Z In Eercises 19 6, find the vlue of the vrible. (See Emple 4.) y S 16 R 1 y E H F G D b ft 1.8 ft 5.8 ft 4.6 ft 3.5 ft 5. z ft 486 hpter 9 Right Tringles nd Trigonometry
24 ERROR NLYSIS In Eercises 7 nd 8, describe nd correct the error in writing n eqution for the given digrm. 7. y z MTHEMTIL ONNETIONS In Eercises 31 34, find the vlue(s) of the vrible(s) b w v z = w (w + v) e g f d 33. y 1 z RESONING Use the digrm. Decide which proportions re true. Select ll tht pply. z 3 4 y h d = f h D MODELING WITH MTHEMTIS In Eercises 9 nd 30, use the digrm. (See Emple 5.) D D = D D = D = D D = D 7. ft 5.5 ft 6 ft 9.5 ft E. 9 E You wnt to determine the height of monument t locl prk. You use crdbord squre to line up the top nd bottom of the monument, s shown t the bove left. Your friend mesures the verticl distnce from the ground to your eye nd the horizontl distnce from you to the monument. pproimte the height of the monument. 30. Your clssmte is stnding on the other side of the monument. She hs piece of rope stked t the bse of the monument. She etends the rope to the crdbord squre she is holding lined up to the top nd bottom of the monument. Use the informtion in the digrm bove to pproimte the height of the monument. Do you get the sme nswer s in Eercise 9? Eplin your resoning. 36. NLYZING RELTIONSHIPS You re designing dimondshped kite. You know tht D = 44.8 centimeters, D = 7 centimeters, nd = 84.8 centimeters. You wnt to use stright crossbr D. bout how long should it be? Eplin your resoning. 37. NLYZING RELTIONSHIPS Use the Geometric Men Theorems (Theorems 9.7 nd 9.8) to find nd D. 0 D D 15 Section 9.3 Similr Right Tringles 487
25 38. HOW DO YOU SEE IT? In which of the following tringles does the Geometric Men (ltitude) Theorem (Theorem 9.7) pply? 40. MKING N RGUMENT Your friend clims the geometric men of 4 nd 9 is 6, nd then lbels the tringle, s shown. Is your friend correct? Eplin 9 4 your resoning. 6 D In Eercises 41 nd 4, use the given sttements to prove the theorem. Given is right tringle. ltitude D is drwn to hypotenuse. 41. PROVING THEOREM Prove the Geometric Men (ltitude) Theorem (Theorem 9.7) by showing tht D = D D. 39. PROVING THEOREM Use the digrm of. opy nd complete the proof of the Pythgoren Theorem (Theorem 9.1). Given In, is right ngle. Prove c = + b STTEMENTS 1. In, is right ngle.. Drw perpendiculr segment (ltitude) from to. RESONS 1.. Perpendiculr Postulte (Postulte 3.) 3. ce = nd cf = b ce + b = + b 4. ddition Property of Equlity 5. ce + cf = + b c(e + f ) = + b e + f = 7. Segment ddition Postulte (Postulte 1.) 8. c c = + b c = + b 9. Simplify. b f D c e 4. PROVING THEOREM Prove the Geometric Men (Leg) Theorem (Theorem 9.8) by showing tht = D nd = D. 43. RITIL THINKING Drw right isosceles tringle nd lbel the two leg lengths. Then drw the ltitude to the hypotenuse nd lbel its length y. Now, use the Right Tringle Similrity Theorem (Theorem 9.6) to drw the three similr tringles from the imge nd lbel ny side length tht is equl to either or y. Wht cn you conclude bout the reltionship between the two smller tringles? Eplin your resoning. 44. THOUGHT PROVOKING The rithmetic men nd geometric men of two nonnegtive numbers nd y re shown. rithmetic men = + y geometric men = y Write n inequlity tht reltes these two mens. Justify your nswer. 45. PROVING THEOREM Prove the Right Tringle Similrity Theorem (Theorem 9.6) by proving three similrity sttements. Given is right tringle. ltitude D is drwn to hypotenuse. Prove D, D, D D Mintining Mthemticl Proficiency Solve the eqution for. (Skills Review Hndbook) = = = 78 Reviewing wht you lerned in previous grdes nd lessons = hpter 9 Right Tringles nd Trigonometry
26 Wht Did You Lern? ore Vocbulry Pythgoren triple, p. 468 geometric men, p. 484 ore oncepts Section 9.1 Theorem 9.1 Pythgoren Theorem, p. 468 ommon Pythgoren Triples nd Some of Their Multiples, p. 468 Theorem 9. onverse of the Pythgoren Theorem, p. 470 Theorem 9.3 Pythgoren Inequlities Theorem, p. 471 Section 9. Theorem Tringle Theorem, p. 476 Theorem Tringle Theorem, p. 477 Section 9.3 Theorem 9.6 Right Tringle Similrity Theorem, p. 48 Theorem 9.7 Geometric Men (ltitude) Theorem, p. 484 Theorem 9.8 Geometric Men (Leg) Theorem, p. 484 Mthemticl Thinking 1. In Eercise 31 on pge 473, describe the steps you took to find the re of the tringle.. In Eercise 3 on pge 480, cn one of the wys be used to show tht ll tringles re similr? Eplin. 3. Eplin why the Geometric Men (ltitude) Theorem (Theorem 9.7) does not pply to three of the tringles in Eercise 38 on pge 488. Study Skills Form Weekly Study Group, Set Up Rules onsider using the following rules. Members must ttend regulrly, be on time, nd prticipte. The sessions will focus on the key mth concepts, not on the needs of one student. Students who skip clsses will not be llowed to prticipte in the study group. Students who keep the group from being productive will be sked to leve the group. 489
27 Quiz Find the vlue of. Tell whether the side lengths form Pythgoren triple. (Section 9.1) Verify tht the segment lengths form tringle. Is the tringle cute, right, or obtuse? (Section 9.1) 4. 4, 3, nd , 9, nd , 15, nd 10 3 Find the vlues of nd y. Write your nswer in simplest form. (Section 9.) y 8. 8 y y 60 Find the geometric men of the two numbers. (Section 9.3) nd nd nd 6 Identify the similr right tringles. Then find the vlue of the vrible. (Section 9.3) D E F 6 H y 9 G 15. J 18 1 z M K L 16. Television sizes re mesured by the length of their digonl. You wnt to purchse television tht is t lest 40 inches. Should you purchse the television shown? Eplin your resoning. (Section 9.1) 17. Ech tringle shown below is right tringle. (Sections ). re ny of the tringles specil right tringles? Eplin your resoning. b. List ll similr tringles, if ny. c. Find the lengths of the ltitudes of tringles nd. 0.5 in. 36 in D E hpter 9 Right Tringles nd Trigonometry
28 9.4 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.9. G.9. The Tngent Rtio Essentil Question How is right tringle used to find the tngent of n cute ngle? Is there unique right tringle tht must be used? Let be right tringle with cute. The tngent of (written s tn ) is defined s follows. length of leg opposite tn = length of leg djcent to = djcent opposite lculting Tngent Rtio Work with prtner. Use dynmic geometry softwre.. onstruct, s shown. onstruct segments perpendiculr to to form right tringles tht shre verte nd re similr to with vertices, s shown K L M N O P Q J I H G F E D Smple Points (0, 0) (8, 6) (8, 0) ngle m = USING PREISE MTHEMTIL LNGUGE To be proficient in mth, you need to epress numericl nswers with degree of precision pproprite for the problem contet. b. lculte ech given rtio to complete the tble for the deciml vlue of tn for ech right tringle. Wht cn you conclude? Rtio tn KD D LE E MF F Using lcultor NG G OH H Work with prtner. Use clcultor tht hs tngent key to clculte the tngent of Do you get the sme result s in Eplortion 1? Eplin. PI I QJ J ommunicte Your nswer 3. Repet Eplortion 1 for with vertices (0, 0), (8, 5), nd (8, 0). onstruct the seven perpendiculr segments so tht not ll of them intersect t integer vlues of. Discuss your results. 4. How is right tringle used to find the tngent of n cute ngle? Is there unique right tringle tht must be used? Section 9.4 The Tngent Rtio 491
29 9.4 Lesson Wht You Will Lern ore Vocbulry trigonometric rtio, p. 49 tngent, p. 49 ngle of elevtion, p. 494 REDING Remember the following bbrevitions. tngent tn opposite opp. djcent dj. Use the tngent rtio. Solve rellife problems involving the tngent rtio. Using the Tngent Rtio trigonometric rtio is rtio of the lengths of two sides in right tringle. ll right tringles with given cute ngle re similr by the Similrity Theorem (Theorem 8.3). So, JKL XYZ, nd you cn write KL rewritten s KL JL = YZ XZ YZ = JL. This cn be XZ, which is trigonometric rtio. So, trigonometric rtios re constnt for given ngle mesure. The tngent rtio is trigonometric rtio for cute ngles tht involves the lengths of the legs of right tringle. ore oncept Tngent Rtio Let be right tringle with cute. The tngent of (written s tn ) is defined s follows. length of leg opposite tn = length of leg djcent to = leg opposite J Y Z K L hypotenuse leg djcent to X USING PREISE MTHEMTIL LNGUGE Unless told otherwise, you should round the vlues of trigonometric rtios to four deciml plces nd round lengths to the nerest tenth. In the right tringle bove, nd re complementry. So, is cute. You cn use the sme digrm to find the tngent of. Notice tht the leg djcent to is the leg opposite nd the leg opposite is the leg djcent to. Finding Tngent Rtios Find tn S nd tn R. Write ech nswer s frction nd s deciml rounded to four plces. opp. S tn S = dj. to S = RT ST = = tn R = opp. R dj. to R = ST RT = = 9 40 = 0.50 S 18 T 8 80 R Monitoring Progress Help in English nd Spnish t igidesmth.com Find tn J nd tn K. Write ech nswer s frction nd s deciml rounded to four plces. 1. K. L 15 J J 3 L K 49 hpter 9 Right Tringles nd Trigonometry
30 Finding Leg Length Find the vlue of. Round your nswer to the nerest tenth. SELETING TOOLS You cn lso use the Tble of Trigonometric Rtios vilble t igidesmth.com to find the deciml pproimtions of trigonometric rtios. Use the tngent of n cute ngle to find leg length. tn 3 = opp. Write rtio for tngent of 3. dj. tn 3 = 11 Substitute. tn 3 = 11 Multiply ech side by. 11 = tn Divide ech side by tn 3. Use clcultor The vlue of is bout STUDY TIP The tngents of ll 60 ngles re the sme constnt rtio. ny right tringle with 60 ngle cn be used to determine this vlue. You cn find the tngent of n cute ngle mesuring 30, 45, or 60 by pplying wht you know bout specil right tringles. Using Specil Right Tringle to Find Tngent Use specil right tringle to find the tngent of 60 ngle. Step 1 ecuse ll tringles re similr, you cn simplify your clcultions by choosing 1 s the length of the shorter leg. Use the Tringle Theorem (Theorem 9.5) to find the length of the longer leg. longer leg = shorter leg Tringle Theorem = 1 3 Substitute. = 3 Simplify Step Find tn 60. tn 60 = opp. dj. tn 60 = 3 1 tn 60 = 3 3 Write rtio for tngent of 60. Substitute. Simplify. The tngent of ny 60 ngle is Monitoring Progress Help in English nd Spnish t igidesmth.com Find the vlue of. Round your nswer to the nerest tenth WHT IF? In Emple 3, the side length of the shorter leg is 5 insted of 1. Show tht the tngent of 60 is still equl to 3. Section 9.4 The Tngent Rtio 493
31 Solving RelLife Problems The ngle tht n upwrd line of sight mkes with horizontl line is clled the ngle of elevtion. Modeling with Mthemtics You re mesuring the height of spruce tree. You stnd 45 feet from the bse of the tree. You mesure the ngle of elevtion from the ground to the top of the tree to be 59. Find the height h of the tree to the nerest foot. h ft ft 1. Understnd the Problem You re given the ngle of elevtion nd the distnce from the tree. You need to find the height of the tree to the nerest foot.. Mke Pln Write trigonometric rtio for the tngent of the ngle of elevtion involving the height h. Then solve for h. 3. Solve the Problem opp. tn 59 = dj. h tn 59 = tn 59 = h Write rtio for tngent of 59. Substitute. Multiply ech side by h Use clcultor. The tree is bout 75 feet tll. 4. Look ck heck your nswer. ecuse 59 is close to 60, the vlue of h should be close to the length of the longer leg of tringle, where the length of the shorter leg is 45 feet. longer leg = shorter leg 3 = Tringle Theorem Substitute. Use clcultor. The vlue of 77.9 feet is close to the vlue of h. Monitoring Progress h in. Help in English nd Spnish t igidesmth.com 6. You re mesuring the height of lmppost. You stnd 40 inches from the bse of in. 494 hpter 9 the lmppost. You mesure the ngle of elevtion from the ground to the top of the lmppost to be 70. Find the height h of the lmppost to the nerest inch. Right Tringles nd Trigonometry
32 9.4 Eercises Tutoril Help in English nd Spnish t igidesmth.com Vocbulry nd ore oncept heck 1. OMPLETE THE SENTENE The tngent rtio compres the length of to the length of.. WRITING Eplin how you know the tngent rtio is constnt for given ngle mesure. Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, find the tngents of the cute ngles in the right tringle. Write ech nswer s frction nd s deciml rounded to four deciml plces. (See Emple 1.) 3. R 8 T 5. G 1 J 5 H S 4. E 7 4 F 5 D 6. J L 3 5 In Eercises 7 10, find the vlue of. Round your nswer to the nerest tenth. (See Emple.) ERROR NLYSIS In Eercises 11 nd 1, describe the error in the sttement of the tngent rtio. orrect the error if possible. Otherwise, write not possible. 6 K tn 55 = 11.0 In Eercises 13 nd 14, use specil right tringle to find the tngent of the given ngle mesure. (See Emple 3.) MODELING WITH MTHEMTIS surveyor is stnding 118 feet from the bse of the Wshington Monument. The surveyor mesures the ngle of elevtion from the ground to the top of the monument to be 78. Find the height h of the Wshington Monument to the nerest foot. (See Emple 4.) 16. MODELING WITH MTHEMTIS Scientists cn mesure the depths of crters on the moon by looking t photos of shdows. The length of the shdow cst by the edge of crter is 500 meters. The ngle of elevtion of the rys of the Sun is 55. Estimte the depth d of the crter. Sun s ry D 37 1 E 35 F tn D = m 17. USING STRUTURE Find the tngent of the smller cute ngle in right tringle with side lengths 5, 1, nd 13. d Section 9.4 The Tngent Rtio 495
33 18. USING STRUTURE Find the tngent of the lrger cute ngle in right tringle with side lengths 3, 4, nd RESONING How does the tngent of n cute ngle in right tringle chnge s the ngle mesure increses? Justify your nswer. 0. RITIL THINKING For wht ngle mesure(s) is the tngent of n cute ngle in right tringle equl to 1? greter thn 1? less thn 1? Justify your nswer. 1. MKING N RGUMENT Your fmily room hs slidingglss door. You wnt to buy n wning for the door tht will be just long enough to keep the Sun out when it is t its highest point in the sky. The ngle of elevtion of the rys of the Sun t this point is 70, nd the height of the door is 8 feet. Your sister clims you cn determine how fr the overhng should etend by multiplying 8 by tn 70. Is your sister correct? Eplin. 4. THOUGHT PROVOKING To crete the digrm below, you begin with n isosceles right tringle with legs 1 unit long. Then the hypotenuse of the first tringle becomes the leg of second tringle, whose remining leg is 1 unit long. ontinue the digrm until you hve constructed n ngle whose tngent is 1. pproimte the mesure of this ngle PROLEM SOLVING Your clss is hving clss picture tken on the lwn. The photogrpher is positioned 14 feet wy from the center of the clss. The photogrpher turns 50 to look t either end of the clss. 1 1 Sun s ry 8 ft ft HOW DO YOU SEE IT? Write epressions for the tngent of ech cute ngle in the right tringle. Eplin how the tngent of one cute ngle is relted to the tngent of the other cute ngle. Wht kind of ngle pir is nd? 3. RESONING Eplin why it is not possible to find the tngent of right ngle or n obtuse ngle. c b Mintining Mthemticl Proficiency Find the vlue of. (Section 9.) Wht is the distnce between the ends of the clss? b. The photogrpher turns nother 10 either wy to see the end of the cmer rnge. If ech student needs feet of spce, bout how mny more students cn fit t the end of ech row? Eplin. 6. PROLEM SOLVING Find the perimeter of the figure, where = 6, D = F, nd D is the midpoint of. E 50 D 35 G Reviewing wht you lerned in previous grdes nd lessons F H 496 hpter 9 Right Tringles nd Trigonometry
34 9.5 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.9. G.9. The Sine nd osine Rtios Essentil Question How is right tringle used to find the sine nd cosine of n cute ngle? Is there unique right tringle tht must be used? Let be right tringle with cute. The sine of nd cosine of (written s sin nd cos, respectively) re defined s follows. length of leg opposite sin = length of hypotenuse cos = = length of leg djcent to = length of hypotenuse hypotenuse djcent opposite lculting Sine nd osine Rtios Work with prtner. Use dynmic geometry softwre.. onstruct, s shown. onstruct segments perpendiculr to to form right tringles tht shre verte nd re similr to with vertices, s shown K L M N O P Q J I H G F E D Smple Points (0, 0) (8, 6) (8, 0) ngle m = b. lculte ech given rtio to complete the tble for the deciml vlues of sin nd cos for ech right tringle. Wht cn you conclude? Sine rtio KD K LE L MF M NG N OH O PI P QJ Q sin NLYZING MTHEMTIL RELTIONSHIPS To be proficient in mth, you need to look closely to discern pttern or structure. osine rtio cos D K E L F M ommunicte Your nswer G N H O I P J Q. How is right tringle used to find the sine nd cosine of n cute ngle? Is there unique right tringle tht must be used? 3. In Eplortion 1, wht is the reltionship between nd in terms of their mesures? Find sin nd cos. How re these two vlues relted to sin nd cos? Eplin why these reltionships eist. Section 9.5 The Sine nd osine Rtios 497
35 9.5 Lesson Wht You Will Lern ore Vocbulry sine, p. 498 cosine, p. 498 ngle of depression, p. 501 Use the sine nd cosine rtios. Find the sine nd cosine of ngle mesures in specil right tringles. Solve rellife problems involving sine nd cosine rtios. Using the Sine nd osine Rtios The sine nd cosine rtios re trigonometric rtios for cute ngles tht involve the lengths of leg nd the hypotenuse of right tringle. ore oncept REDING Remember the following bbrevitions. sine sin cosine cos hypotenuse hyp. Sine nd osine Rtios Let be right tringle with cute. The sine of nd cosine of (written s sin nd cos ) re defined s follows. length of leg opposite sin = length of hypotenuse length of leg djcent to cos = length of hypotenuse = = leg opposite hypotenuse leg djcent to Finding Sine nd osine Rtios Find sin S, sin R, cos S, nd cos R. Write ech nswer s frction nd s deciml rounded to four plces. opp. S sin S = hyp. = RT SR = opp. R sin R = = ST hyp. SR = dj. to S cos S = = ST hyp. SR = 16 dj. to R 0.46 cos R = 65 hyp. R S 16 T = RT SR = In Emple 1, notice tht sin S = cos R nd sin R = cos S. This is true becuse the side opposite S is djcent to R nd the side opposite R is djcent to S. The reltionship between the sine nd cosine of S nd R is true for ll complementry ngles. ore oncept Sine nd osine of omplementry ngles The sine of n cute ngle is equl to the cosine of its complement. The cosine of n cute ngle is equl to the sine of its complement. Let nd be complementry ngles. Then the following sttements re true. sin = cos(90 ) = cos sin = cos(90 ) = cos cos = sin(90 ) = sin cos = sin(90 ) = sin 498 hpter 9 Right Tringles nd Trigonometry
Geometry 71 Geometric Mean and the Pythagorean Theorem
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