8-1 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted Word: hypotenuse Definition: In right tringle, the sides tht form the right ngle re the legs. Min Ide: The legs of right tringle re perpendiulr. The hypotenuse is the side opposite the right ngle. Use Your Voulry 2. Underline the orret word to omplete the sentene. The hypotenuse is the longest / shortest side in right tringle. Write T for true or F for flse. F T T 3. The hypotenuse of right tringle n e ny one of the three sides. 4. One leg of the tringle t the right hs length 9 m.. The hypotenuse of the tringle t the right hs length 1 m. leg hypotenuse leg 1 m m 9 m Chpter 8 202
Theorems 8-1 nd 8-2 Pythgoren Theorem nd Its Converse Pythgoren Theorem If tringle is right tringle, then the sum of the squres of the lengths of the legs is equl to the squre of the length of the hypotenuse. If nabc is right tringle, then 2 1 2 2. Converse of the Pythgoren Theorem If the sum of the squres of the lengths of two sides of tringle is equl to the squre of the length of the third side, then the tringle is right tringle. If 2 1 2 2, then nabc is right tringle. 6. Cirle the eqution tht shows the orret reltionship mong the lengths of the legs nd the hypotenuse of right tringle. 2 1 2 2 2 1 2 2 2 1 2 2 Underline the orret words to omplete eh sentene. 7. A tringle with side lengths 3, 4, nd is / is not right tringle euse 3 2 1 4 2 is equl / not equl to 2. 8. A tringle with side lengths 4,, nd 6 is / is not right tringle euse 4 2 1 2 is equl / not equl to 6 2. A B C Prolem 1 Finding the Length of the Hypotenuse Got It? The legs of right tringle hve lengths 10 nd 24. Wht is the length of the hypotenuse? 9. Lel the tringle t the right. 10. Use the justifitions elow to find the length of the hypotenuse. 2 1 2 2 Pythgoren Theorem 10 2 2 1 24 2 Sustitute for nd. 100 1 76 2 Simplify. 676 26 2 Add. Tke the positive squre root. 11. The length of the hypotenuse is 26.. One Pythgoren triple is,, nd. If you multiply eh numer y 2, wht numers result? How do the numers tht result ompre to the lengths of the sides of the tringle in Exerises 9 11? 10, 24, 26. Answers my vry. Smple: The numers re the sme 10 24 s the lengths of the sides of the tringle in Exerises 9 11. 203 Lesson 8-1
Prolem 3 Finding Distne Got It? The size of omputer monitor is the length of its digonl. You wnt to uy 19-in. monitor tht hs height of 11 in. Wht is the width of the monitor? Round to the nerest tenth of n inh. 19 in. 11 in.. Lel the digrm of the omputer monitor t the right. 14. The eqution is solved elow. Write justifition for eh step. in. 2 1 2 2 Pythgoren Theorem 11 2 1 2 19 2 Sustitute. 1 1 2 361 Simplify. 1 2 1 1 2 361 2 1 2 240 Sutrt 1 from eh side. Simplify. "240 Tke the positive squre root. < 1.49193338 Use lultor. 1. To the nerest tenth of n inh, the width of the monitor is 1. in. Prolem 4 Identifying Right Tringle Got It? A tringle hs side lengths 16, 48, nd 0. Is the tringle right tringle? Explin. 16. Cirle the eqution you will use to determine whether the tringle is right tringle. 16 2 1 48 2 0 0 2 16 2 1 0 2 0 48 2 48 2 1 0 2 0 16 2 17. Simplify your eqution from Exerise 16. 16 2 1 48 2 0 0 2 26 1 2304 0 200 260 u 200 18. Underline the orret words to omplete the sentene. The eqution is true / flse, so the tringle is / is not right tringle. A Pythgoren triple is set of nonzero whole numers,, nd tht stisfy the eqution 2 1 2 2. If you multiply eh numer in Pythgoren triple y the sme whole numer, the three numers tht result lso form Pythgoren triple. Chpter 8 204
Theorems 8-3 nd 8-4 Pythgoren Inequlity Theorems Theorem 8-3 If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the lengths of the other two sides, then the tringle is otuse. Theorem 8-4 If the squre of the length of the longest side of tringle is less thn the sum of the squres of the lengths of the other two sides, then the tringle is ute. Use the figures t the right. Complete eh sentene with ute or otuse. 19. In nabc, 2. 2 1 2, so nabc is 9. 20. In nrst, s 2, r 2 1 t 2, so nrst is 9. otuse ute A R t C S r s T B Lesson Chek Do you UNDERSTAND? Error Anlysis A tringle hs side lengths 16, 34, nd 30. Your friend sys it is not right tringle. Look t your friend s work nd desrie the error. 21. Underline the length tht your friend used s the longest side. Cirle the length of the longest side of the tringle. 16 2 + 34 2? = 30 2? 26 + 116 = 900 14 900 16 30 34 22. Write the omprison tht your friend should hve used to determine whether the tringle is right tringle. 16 2 1 30 2 0 34 2 23. Desrie the error in your friend s work. Answers my vry. Smple: My friend used the wrong length for in the omprison. The omprison should e 16 2 1 30 2 0 34 2. Mth Suess Chek off the voulry words tht you understnd. hypotenuse leg Pythgoren Theorem Pythgoren triple Rte how well you n use the Pythgoren Theorem nd its onverse. Need to review 0 2 4 6 8 10 Now I get it! 20 Lesson 8-1
8-2 Speil Right Tringles Voulry Review 1. Cirle the segment tht is digonl of squre ABCD. AB AC AD BC CD 2. Underline the orret word to omplete the sentene. A digonl is line segment tht joins two sides / verties of polygon. Voulry Builder D A C B omplement (noun) KAHM pluh munt Other Word Form: omplementry (djetive) Mth Usge: When the mesures of two ngles hve sum of 90, eh ngle is omplement of the other. Nonexmple: Two ngles whose mesures sum to 180 re supplementry. Use Your Voulry Complete eh sttement with the word omplement or omplementry. 3. If m/a 40 nd m/b 0, the ngles re 9. omplementry 4. If m/a 30 nd m/b 60, /B is the 9 of /A. omplement. /P nd /Q re 9 euse the sum of their mesures is 90. omplementry Complete. 6. If /R hs mesure of 3, then the omplement of /R hs mesure of. 7. If /X hs mesure of 22, then the omplement of /X hs mesure of 68. 8. If /C hs mesure of 6, then the omplement of /C hs mesure of 2. 9. Cirle the omplementry ngles. 60 40 0 0 Chpter 8 206
Theorem 8-4 -4-90 Tringle Theorem In 48-48-908 tringle, oth legs re ongruent nd the length of the hypotenuse is "2 times the length of leg. s 2 4 s Complete eh sttement for 48 248 2908 tringle. 10. hypotenuse "2? leg 11. If leg 10, then hypotenuse "2? 10. 4 s Prolem 1 Finding the Length of the Hypotenuse Got It? Wht is the length of the hypotenuse of 48-48-908 tringle with leg length!3?. Use the justifitions to find the length of the hypotenuse. hypotenuse "2? leg 48-48-908 Tringle Theorem "2? "3 Sustitute. "2? "3 Commuttive Property of Multiplition. "6 Simplify. Prolem 2 Finding the Length of Leg Got It? The length of the hypotenuse of 48-48-908 tringle is 10. Wht is the length of one leg?. Will the length of the leg e greter thn or less thn 10? Explin. Less thn. Explntions my vry. Smple: The hypotenuse is the longest side. 14. Use the justifitions to find the length of one leg. hypotenuse "2? leg 10 "2? leg Sustitute. 48-48-908 Tringle Theorem 10 "2? leg Divide eh side y "2. "2 "2 10 leg "2 Simplify. 10 "2 leg? "2 "2 Multiply y form of 1 to rtionlize the denomintor. 10"2 leg 2 Simplify. leg "2 Divide y 2. 207 Lesson 8-2
Prolem 3 Finding Distne Got It? You pln to uild pth long one digonl of 100 ft-y- 100 ft squre grden. To the nerest foot, how long will the pth e? 1. Use the words pth, height, nd width to omplete the digrm. 16. Write L for leg or H for hypotenuse to identify eh prt of the right tringle in the digrm. H pth L height L width 17. Sustitute for hypotenuse nd leg. Let h the length of the hypotenuse. hypotenuse "2? leg h "2? 100 18. Solve the eqution. Use lultor to find the length of the pth. h!2? 100 h N 141.4262 height width pth 19. To the nerest foot, the length of the pth will e 141 feet. Theorem 8-6 30-60 -90 Tringle Theorem In 308-608-908 tringle, the length of the hypotenuse is twie the length of the shorter leg. The length of the longer leg is "3 times the length of the shorter leg. Complete eh sttement for 308-608-908 tringle. 20. hypotenuse 2? shorter leg 21. longer leg "3? shorter leg Prolem 4 Think f is the length of the hypotenuse. I n write n eqution relting the hypotenuse nd the shorter leg 3 3 Now I n solve for f. Using the Length of One Side Got It? Wht is the vlue of f in simplest rdil form? 22. Complete the resoning model elow. of the 30-60 -90 tringle. hypotenuse f f Write 2 2 shorter leg 10 3 3 3 3 œ3 3 60 2s 30 60 s 30 f s V3 Chpter 8 208
Prolem Applying the 30-60 -90 Tringle Theorem Got It? Jewelry Mking An rtisn mkes pendnts in the shpe of equilterl tringles. Suppose the sides of pendnt re 18 mm long. Wht is the height of the pendnt to the nerest tenth of millimeter? 18 mm 18 mm 23. Cirle the formul you n use to find the height of the pendnt. hypotenuse 2? shorter leg 24. Find the height of the pendnt. longer leg!3? shorter leg 18 mm longer leg "3? shorter leg "3? 9 N 1.884727 2. To the nerest tenth of millimeter, the height of the pendnt is 1.6 mm. Lesson Chek Do you UNDERSTAND? Resoning A test question sks you to find two side lengths of 4-4 -90 tringle. You know tht the length of one leg is 6, ut you forgot the speil formul for 4-4 -90 tringles. Explin how you n still determine the other side lengths. Wht re the other side lengths? 26. Underline the orret word(s) to omplete the sentene. In 4-4 -90 tringle, the lengths of the legs re different / the sme. 27. Use the Pythgoren Theorem to find the length of the longest side. 28. The other two side lengths re 6 nd 6"2. Mth Suess Chek off the voulry words tht you understnd. leg hypotenuse right tringle Pythgoren Theorem Rte how well you n use the properties of speil right tringles. Need to review longest side: 2 6 2 1 6 2 2 36 1 36 2 72 "72 6"2 0 2 4 6 8 10 Now I get it! 209 Lesson 8-2
8-3 Trigonometry Voulry Review The Venn digrm t the right shows the reltionship etween similr nd ongruent figures. Write T for true or F for flse. F T 1. All similr figures re ongruent figures. 2. All ongruent figures re similr figures. Similr Figures Congruent Figures T 3. Some similr figures re ongruent figures. 4. Cirle the postulte or theorem you n use to verify tht the tringles t the right re similr. AA, Postulte SAS, Theorem SSS, Theorem Voulry Builder rtio (noun) RAY shee oh Relted Words: rte, rtionl Definition: A rtio is the omprison of two quntities y division. Exmple: If there re 6 tringles nd squres, the rtio of tringles to squres is 6 nd the rtio of squres to tringles is 6. Use Your Voulry Use the tringle t the right for Exerises 7.. Cirle the rtio of the length of the longer leg to the length of the shorter leg. 6. Cirle the rtio of the length of the shorter leg to the length of the hypotenuse. 7. Cirle the rtio of the length of the longer leg to the length of the hypotenuse. Chpter 8 210
Key Conept The Trigonometri Rtios sine of /A osine of /A tngent of /A length of leg opposite/a length of hypotenuse length of leg djent to/a length of hypotenuse length of leg opposite/a length of leg djent to/a A B C Drw line from eh trigonometri rtio in Column A to its orresponding rtio in Column B. Column A 8. sin B 9. os B 10. tn B Column B 11. Resoning Suppose nabc is right isoseles tringle. Wht would the tngent of /B equl? Explin. Explntions my vry. Smple: 1. The legs would e ongruent, so would equl 1. Prolem 1 Writing Trigonometri Rtios Got It? Wht re the sine, osine, nd tngent rtios for lg?. Cirle the mesure of the leg opposite /G. 8 1 17. Cirle the mesure of the hypotenuse. 8 1 17 14. Cirle the mesure of the leg djent to /G. 8 1 17 1. Write eh trigonometri rtio. sin G os G opposite hypotenuse djent hypotenuse tn G opposite djent 1 8 1 17 8 17 T 17 1 G 8 R 211 Lesson 8-3
Prolem 2 Using Trigonometri Rtio to Find Distne Got It? Find the vlue of w to the nerest tenth. Below is one student s solution. 17 4 w os 4 w 17 os 4 (17) w 9.992349289 w 10 w 16. Cirle the trigonometri rtio tht uses sides w nd 17. sin 48 os 48 tn 48 17. Wht error did the student mke? Answers my vry. Smple: The student wrote os 4 w 17 rther thn sin 4 w 17. 18. Find the vlue of w orretly. sin 4 w 17 sin 4 (17) w.732889 N w.8 N w 19. The vlue of w to the nerest tenth is.8. Prolem 3 Using Inverses Got It? Use the figure elow. Wht is mly to the nerest degree? P 100 T 41 Y 20. Cirle the lengths tht you know. hypotenuse side djent to /Y side opposite /Y 21. Cross out the rtios tht you will NOT use to find m/y. sine osine tngent 22. Underline the orret word to omplete the sttement. If you know the sine, osine, or tngent rtio of n ngle, you n use the inverse / rtio to find the mesure of the ngle. Chpter 8 2
23. Follow the steps to find m/y. 1 Write the rtio. 100 tn Y 41 2 Use the inverse. 100 Y tn ( 1 41 ) 3 Use lultor. Y 67.70637084 24. To the nerest degree, m/y < 68. Lesson Chek Do you UNDERSTAND? Error Anlysis A student sttes tht sin A S sin X euse the lengths of the sides of kabc re greter thn the lengths of the sides of kxyz. Wht is the student s error? Explin. Y B Underline the orret word(s) to omplete eh sentene. 2. nabc nd nxyz re / re not similr. Z 3 X C 3 A 26. /A nd /X re / re not ongruent, so sin 38 is / is not equl to sin 38. 27. Wht is the student s error? Explin. Answers my vry. Smple: The student did not look t the mesures of la nd lx. Congruent ngles hve equl sine rtios. Mth Suess Chek off the voulry words tht you understnd. trigonometri rtios sine osine tngent Rte how well you n use trigonometri rtios. Need to review 0 2 4 6 8 10 Now I get it! 2 Lesson 8-3
8-4 Angles of Elevtion nd Depression Voulry Review Underline the orret word(s) or numer to omplete eh sentene. 1. The mesure of right ngle is greter / less thn the mesure of n ute ngle nd greter / less thn the mesure of n otuse ngle. 2. A right ngle hs mesure of 4 / 90 /180. 3. Lines tht interset to form four right ngles re prllel / perpendiulr lines. 4. Cirle the right ngle(s) in the figure. /ACB /ADB /BAC A /BAD /CBA /DBA Voulry Builder D B C elevtion (noun) el uh VAY shun Relted Word: depression Definition: The elevtion of n ojet is its height ove given level, suh s eye level or se level. Mth Usge: Angles of elevtion nd depression re ute ngles of right tringles formed y horizontl distne nd vertil height. Use Your Voulry Complete eh sttement with the orret word from the list elow. Use eh word only one. elevte elevted elevtion. John 9 his feet on footstool. 6. The 9 of Mt MKinley is 20,320 ft. 7. You 9 n ojet y rising it to higher position. elevted elevtion elevte Chpter 8 214
Prolem 1 Identifying Angles of Elevtion nd Depression Got It? Wht is desription of l2 s it reltes to the sitution shown? Write T for true or F for flse. T 8. /2 is ove the horizontl line. T F 9. /2 is the ngle of elevtion from the person in the hot-ir lloon to the ird. 10. /2 is the ngle of depression from the person in the hot-ir lloon to the ird. F 11. /2 is the ngle of elevtion from the top of the mountin to the person in the hot-ir lloon.. Desrie /2 s it reltes to the sitution shown. Answers my vry. Smple: l2 is the ngle of elevtion from the person in the hot-ir lloon to the ird. Prolem 2 Using the Angle of Elevtion Got It? You sight rok limer on liff t 32 ngle of elevtion. Your eye level is 6 ft ove the ground nd you re 1000 feet from the se of the liff. Wht is the pproximte height of the rok limer from the ground?. Use the informtion in the prolem to omplete the prolem-solving model elow. Eye level 32 1000 ft Know Need Pln Angle of elevtion Height of limer from Find the length of the is 32 8. the ground leg opposite 328 y using tn 32 8. Distne to the liff is 1000 ft. Then dd 6 ft. Eye level is 6 ove the ground. ft Climer 21 Lesson 8-4
14. Explin why you use tn 328 nd not sin 328 or os 328. Answers my vry. Smple: The sine rtio involves two unknowns. The osine rtio involves the hypotenuse nd 1000, ut I do not wnt to know the hypotenuse. The rtio tht uses the unknown height nd 1000 is the tngent rtio. 1. The prolem is solved elow. Use one of the resons from the list t the right to justify eh step. tn 328 d 1000 Write the eqution. Solve for d. Use lultor. Write the eqution. (tn 328) 1000 d Solve for d. d < 624.869319 Use lultor. 16. The height from your eye level to the limer is out 62 ft. 17. The height of the rok limer from the ground is out 631 ft. Prolem 3 Using the Angle of Depression Got It? An irplne pilot sights life rft t 26 ngle of depression. The irplne s ltitude is 3 km. Wht is the irplne s horizontl distne d from the rft? 18. Lel the digrm elow. ltitude 3 km Not to sle 26º Angle of elevtion 26º d Angle of depression horizontl distne Rft 19. Cirle the eqution you ould use to find the horizontl distne d. sin 268 3 d 20. Solve your eqution from Exerise 19. tn 268 3 d d 3 tn 268 d 6.1091 os 268 3 d tn 268 3 d 21. To the nerest tenth, the irplne s horizontl distne from the rft is 6.2 km. Chpter 8 216
Lesson Chek Do you UNDERSTAND? Voulry How is n ngle of elevtion formed? Underline the orret word(s) to omplete eh sentene. 22. The ngle of elevtion is formed ove / elow horizontl line. 23. The ngle of depression is formed ove / elow horizontl line. 24. The mesure of n ngle of elevtion is equl to / greter thn / less thn the mesure of the ngle of depression. Lesson Chek Do you UNDERSTAND? Error Anlysis A homework question sys tht the ngle of depression from the ottom of house window to ll on the ground is 20. At the right is your friend s sketh of the sitution. Desrie your friend s error. 2. Is the ngle tht your friend identified s the ngle of depression formed y the horizontl nd the line of sight? Yes / No 26. Is the orret ngle of depression djent to or opposite the ngle identified y your friend? 27. Desrie your friend s error. Mth Suess Chek off the voulry words tht you understnd. ngle of elevtion ngle of depression trigonometri rtios Rte how well you n use ngles of elevtion nd depression. Need to review 0 2 4 6 8 10 Now I get it! 20 djent to / opposite Answers my vry. Smple: My friend identified the wrong ngle. The orret ngle of depression is elow the horizontl line. 217 Lesson 8-4
Lw of Sines 8- Voulry Review 1. Drw line segment from eh ngle of the tringle to its opposite side. C A B 2. Cirle the orret word. A rtio is the omprison of two quntities y ddition sutrtion multiplition division sine (noun) syn Relted Words: tringle, side length, ngle mesure, opposite, osine Definition: In right tringle, sine is the rtio of the side opposite given ute ngle to the hypotenuse. Exmple: If you know the mesure of n ute ngle of right tringle nd the length of the opposite side, you n use the sine rtio to find the length of the hypotenuse. Use Your Voulry 3. A tringle hs given ute ngle. Cirle its sine rtio. djent hypotenuse hypotenuse opposite opposite hypotenuse opposite djent 4. A right tringle hs one ute ngle mesuring 36.98. The length of the side djent to this ngle is 4 units, nd the length of the side opposite this ngle is 3 units. The length of the hypotenuse is units. Cirle the sine rtio of the 36.98 ngle. 43 Chpter 8 3 4 4 3 218 Voulry Builder
Lw of Sines For ny nabc, let the lengths of the sides opposite ngles A, B, nd C e,, nd, respetively. Then the Lw of Sines reltes the sine of eh ngle to the length of its opposite side. sin A sin B sin C. If you know 2 ngles nd 1 side of tringle, n you find ll of the missing mesures? Explin. Yes; sine the sum of tringle s ngles re 1808, you n find the third ngle; then use the Lw of Sines to find the other 2 sides. A C B Prolem 1 Using the Lw of Sines (AAS) Got It? In DABC, m/a 48, m/b 93, nd AC 1. Wht is AB to the nerest tenth? 6. Find nd lel m/c. 1808 488 1 938 1 398 C 7. Lel side lengths,, nd. Whih side is the length of AB? 8. Cirle the eqution whih n e used to solve this prolem. Explin your resoning. sin C sin A sin C sin B sin B sin A 39 93 B 1 48 A Answers my vry. Smple: I wnt to pik the eqution in whih I hve 3 out of the 4 vlues. 9. Reple the vriles in the eqution with vlues from DABC. Prolem 2 sin 398 sin 938 1 10. Find the sine vlues of the given ngles, ross multiply, then solve for. < (.6293 ) ( 1 ) < (.9986 ) 11. The length of AB is out 9. units. 9.43 Using the Lw of Sines (SSA) Got It? In DKLM, LM 9, KM 14, nd m/l 10. To the nerest tenth, wht is m/k? m L 10 9 k. Lel the tringle with informtion from the prolem nd the length of the sides s k, l, m. K 14 l M 219 Lesson 8-
. Use the letter tht represents the length of KM to write pir of rtios using some of the letters k, l, m, K, L nd M. sin L sin K l k 14. Fill in the vlues in the eqution from Exerise nd solve for sin K. sin 10 14 sin K 9 sin K < (.969)(9) 14 <.6210 sin K <.6210 1. Use your lultor nd tke the inverse sine of oth sides of the eqution to find m/k. sin 2 1 (sin K) < sin 21.6210, therefore m/k < 38.4 Prolem 3 Using the Lw of Sines to Solve Prolem Got It? The right-fielder fields softll etween first se nd seond se s shown in the figure. If the right-fielder throws the ll to seond se, how fr does she throw the ll? 16. Underline the orret word to omplete eh sentene. In this prolem, the solution is side / ngle. To find the solution, I need to first find missing side / ngle. 17. In order to use the Lw of Sines wht informtion will you need tht is missing nd why? I need to find the mesure of the ngle formed y seond se, the right fielder, nd first se euse I need the mesure of the ngle opposite the given side to use the Lw of Sines. 18. Cirle the eqution you ould use to solve for the missing solution. sin 728 60 sin 408 sin 728 60 sin 688 sin 688 60 sin 728 19. Fill in the lnks to omplete the eqution. Then solve the eqution nd find the solution. sin 728 sin 408 60 2nd Bse 0.911 0.6428 (0.911) < 60(0.6428) < 40.6 60 68 60 ft Right-fielder 72 40 1st Bse Kimmy throws the ll out 40.6 feet. Chpter 8 220
Lesson Chek Do you UNDERSTAND? Resoning If you know the three side lengths of tringle, n you use the Lw of Sines to find the missing ngle mesures? Explin. 20. Wht do AAS, ASA, nd SSA stnd for? Mth eh term with its definition. Then tell wht the three terms hve in ommon. AAS Side-Side-Angle ASA Angle-Angle-Side SSA Angle-Side-Angle They ll inlude sides nd ngles. 21. If you know only the three side lengths of tringle, n you use the Lw of Sines to find the missing ngle mesures? Explin. Answers my vry. Smple: No; To use the Lw of Sines, you need 2 ngles nd 1 side or 2 sides nd 1 ngle. Error Anlysis In DPQR, PQ 4 m, QR 3 m, nd m/r 7. Your friend uses the Lw of Sines to write Explin the error. sin 78 sin x8 4 to find m/q. 3 P 22. Lel the digrm with the given informtion. Did your friend orretly mth the ngles nd the sides? 4 Answers my vry. Smple: The side length opposite /R is 4, _ not 3. /Q is not opposide side length 4. _ 7 R x 3 Q Mth Suess Chek off the voulry words tht you understnd. Lw of Sines rtio djent inverse sine Rte how well you n use the Lw of Sines. Need to review 0 2 4 6 8 Now I get it! 10 221 Lesson 8-
Lw of Cosines 8-6 Voulry Review Look t DABC. C A B nd 1. Nme the sides tht re djent to ngle A. 2. Whih side is opposite of ngle B? 3. Identify eh ngle mesure s ute, right, or otuse. 48 ute 1008 otuse 908 right Cosine (noun) KOH syn Relted Word: tringle, side length, ngle mesure, opposite, sine Definition: In right tringle, osine is the rtio of the side djent to given ute ngle to the hypotenuse. Exmple: If you know the mesure of n ute ngle of right tringle nd the length of the djent side, you n use the osine rtio to find the length of the hypotenuse. Use Your Voulry 4. A tringle hs given ute ngle. Cirle its osine rtio. djent hypotenuse hypotenuse djent opposite hypotenuse djent opposite. A right tringle hs one ute ngle mesuring 3.18, the length of the side djent to this ngle is 9 units, nd the length of the side opposite this ngle is units. The length of the hypotenuse is 1 units. Cirle the osine rtio of the 3.18 ngle. 9 Chpter 8 1 9 1 1 9 1 222 Voulry Builder
Lw of Cosines For ny nabc with side lengths,, nd opposite ngles A, B, nd C, respetively, the Lw of Cosines reltes the mesures of the tringles ording to the following equtions. 2 2 1 2 2 2 os A C 2 2 1 2 2 2 os B 2 2 1 2 2 2 os C A 6. Cirle the eqution tht is true for DDEF. d 2 f 2 1 e 2 2 2de os D E f 2 d 2 1 e 2 2 2de os F d f e 2 d 2 1 f 2 2 df os E F e B D Prolem 1 Using the Lw of Cosines (SAS) Got It? In DLMN, m/l 1048, LM 48, nd LN 29. Find MN to the nerest tenth. 7. Lel the sides of DLMN with the letters l, m, nd n. 8. Use the informtion in the prolem to omplete the prolem-solving model elow. M n 48 l 104 L 29 N m Know LM is opposite / LM 48 letter LN is opposite / LN 29 letter 9. Find MN y solving for l.. I 2 m 2 1 n 2 2 2( m )( n ) os /L. Write n eqution using l, m, n, nd /L.. l 2 29 2 1 48 2 2 2( 29 )( 48 ) os 104. Sustitute the vlues from the tringle.. l 2 < 841 1 2304 2 (2784? 20.242). Use the Order of Opertions nd l 2 l 2 < N M n m < 314 1 673.728 3818.728 Need MN letter An eqution using letters l, m, n, nd /L. I solve for l 2. Pln Beuse you know m/ L nd need MN, sustitute the ngle mesure nd the two side lengths into the eqution nd solve for l. d. l < 61.8 MN d. Tke the squre root of oth sides. 223 Lesson 8-6
Prolem 2 Using the Lw of Cosines (SSS) Got It? In DTUV ove, find m/t to the nerest tenth of degree. 10. Lel the sides of the tringle with t, u, nd r. 11. Solve for m/t following the given STEPS. t 2 u 2 1 v 2 2 2( u )( v ) os T Write n eqution using the Lw of Cosines. 7.1 2 6.7 2 1 4.4 2 2 2( 6.7 )( 4.4 ) os T Sustitute the vlues from the tringle. 0.41 44.89 1 19.36 2 8.96 os T Simplify y squring nd multiplying. 0.41 64.2 2 8.96 os T Add the first two numers. U 4.4 v T t u 7.1 6.7 V 0.41 2 64.2 2 8.96 os T Get oeffiient of os T nd os T lone. 2.84 8.96 os T Divide y the oeffiient of os T. os 21 0.23473 < T Tke the inverse osine of oth sides of the eqution. m/t < 76.4 Prolem 3 Using the Lw of Cosines to Solve Prolem Got It? You nd friend hike 1.4 miles due west from mpsite. At the sme time two other friends hike 1.9 miles t heding of S 118W (118west of south) from the mpsite. To the nerest tenth of mile, how fr prt re the two groups?. Lel the model with informtion from the prolem nd letter the ngles nd sides. West = 1.4 miles = 1.9 miles 11 mpsite South. Find the mesure of the ngle tht is the omplement of the 118ngle. 908 2 118 798 Chpter 8 224
14. Write nd solve n eqution for finding the distne etween the two groups. 2 1.42 1 1.92 2 2(1.4)(1.9) os(79) 2 1.96 1 3.61 2.32 (.191) 2.7 2 1.02 2 4. < 2. miles Lesson Chek Do you UNDERSTAND? Writing Explin how you hoose etween the Lw of Sines nd the Lw of Cosines when finding the mesure of missing ngle or side. 1. Write C if you would use the Lw of Cosines to find missing mesure in tringle or S if you would use the Lw of Sines. C The lengths of two sides nd the mesure of the inluded ngle re given. Find the length of the third side. C S The lengths of three sides re given. Find the mesure of one ngle. The mesures of two ngles nd the length of the inluded side re given. Find the length of nother side. 16. Explin how to hoose etween the Lw of Sines nd the Lw of Cosines in solving tringle. Answers my vry. Smple: Use the Lw of Cosines when you know SAS _ or SSS. Use the Lw of Sines when you know SSA or ASA. _ Mth Suess Chek off the voulry words tht you understnd. Lw of Cosines Lw of Sines trigonometry Rte how well you n use the Lw of Cosines. Need to review 0 2 4 6 8 Now I get it! 10 22 Lesson 8-6