Geometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 4 / 240. Slide 3 / 240. Slide 6 / 240.

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Martha Mason
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1 Slide 1 / 240 New Jersey enter for Tehing nd Lerning Progressive Mthemtis Inititive This mteril is mde freely ville t nd is intended for the non-ommeril use of students nd tehers. These mterils my not e used for ny ommeril purpose without the written permission of the owners. NJTL mintins its wesite for the onveniene of tehers who wish to mke their work ville to other tehers, prtiipte in virtul professionl lerning ommunity, nd/or provide ess to ourse mterils to prents, students nd others. Slide 2 / 240 Geometry Trigonometry of Right Tringles lik to go to wesite: Slide 3 / 240 Slide 4 / 240 Tle of ontents Pythgoren Theorem Similrity in Right Tringles Speil Right Tringles Trigonometri Rtios lik on Topi to go to tht setion Pythgoren Theorem Solving Right Tringles ngles of Elevtion nd epression Lw of Sines nd Lw of osines re of n Olique Tringle Return to the Tle of ontents Slide 5 / 240 Slide 6 / 240 efore lerning out similr right tringles nd trigonometry, we need to review the Pythgoren Theorem nd the Pythgoren Theorem onverse. Rell tht right tringle is tringle with right ngle. leg hypotenuse leg The sides form tht right ngle re the legs. The side opposite the right ngle is the hypotenuse. The hypotenuse is lso the longest side.

2 Slide 7 / 240 Pythgoren Theorem In right tringle, the sum of the squres of the lengths of the legs is equl to the squre of the length of the hypotenuse. Emple: Slide 8 / 240 Find the length of the missing side of the right tringle. 9 leg 2 + leg 2 = hypotenuse 2 or = 2 Is the missing side leg or the hypotenuse of the right tringle? Slide 9 / 240 Slide 10 / 240 Solve for : Emple: = = = 2 # 15 = 9 Find the length of the missing side of the right tringle. Is the missing side leg or the hypotenuse of the right tringle? is etrneous solution, distne n not equl negtive numer. = 15 Slide 11 / The missing side is the of the right tringle. leg hypotenuse 6 9 Slide / Find the length of the missing side. 6 9

Slide 13 / 240 3 The missing side is the of the right tringle. Slide 14 / 240 4 Find the length of the missing side.

3 Slide 13 / The missing side is the of the right tringle. Slide 14 / Find the length of the missing side. leg hypotenuse Slide 15 / 240 Slide 16 / 240 Rel World pplition The sfe distne of the se of the ldder from wll it lens ginst should e one-fourth of the length of the ldder Solve using + =? 28 feet Thus, the ottom of 28-foot ldder should e 7 feet from the wll. How fr up the wll will ldder reh?? 28 feet 7 feet 7 feet The ldder will reh feet up the wll sfely. Slide 17 / 240 Rel World pplition 50 The dimensions of high shool sketll ourt re 84' long nd 50' wide. Wht is the length from one orner of the ourt to the opposite orner? 84 Slide 18 / N ourt is 50 feet wide nd the length from one orner of the ourt to the opposite orner is feet. How long is the ourt? (Round the nswer to the nerest whole numer) feet feet 118 feet 94 feet

4 Slide 19 / 240 Pythgoren Theorem pplitions Slide 20 / 240 Emple Find the perimeter of the squre. The Pythgoren Theorem n lso e used in figures tht ontin right ngles. 18 m Psq = 4s note: efore finding the perimeter of the squre, we need to first find the length of eh side. Slide 21 / 240 Slide 22 / 240 Rememer, in squre ll sides re ongruent. Strt here: 18 m = 18 2 Emple Find the re of the tringle. The se of the tringle is given, ut we need to find the height of the tringle. 1 = 2 h 13 feet 13 feet 10 feet Slide 23 / 240 y definition, the ltitude (or height) of n isoseles tringle is the perpendiulr isetor of the se. Slide 24 / 240 Try this... Find the perimeter of the retngle. 8 in Pret = 2l + 2w 13 feet h 13 feet 10 in 5 feet 5 feet

5 Slide 25 / Find the re of the retngle. 0 squre feet 84 squre feet 46 squre inhes 46 squre feet 8 feet 17 feet Slide 26 / Find the perimeter of the squre. (Round to the nerest tenth).8 m 25.5 m 25.6 m 36 m 9 m Slide 27 / Find the re of the tringle. Slide 28 / Find the re of the tringle. 7 inhes 7 inhes 7 inhes 7 inhes 10 inhes 4 inhes Slide 29 / 240 onverse of the Pythgoren Theorem If the squre of the longest side of tringle is equl to the sum of the squres of the other two sides, then the tringle is right tringle. If 2 = 2 + 2, then is right tringle. Slide 30 / 240 Emple Tell whether the tringle is right tringle. Rememer is the longest side F E 7

6 Slide 31 / 240 Theorem If the squre of the longest side of tringle is greter thn the sum of the squres of the other two sides, then the tringle is otuse. If 2 > 2 + 2, then is otuse. Slide 32 / 240 If the squre of the longest side of tringle is less thn the sum of the squres of the other two sides, then the tringle is ute. If 2 < 2 + 2, then is ute. Theorem Slide 33 / 240 Emple lssify the tringle s ute, right, or otuse. Slide 34 / lssify the tringle s ute, right, otuse, or not tringle ute right 17 otuse not tringle Slide 35 / 240 Slide 36 / lssify the tringle s ute, right, otuse, or not tringle. lssify the tringle s ute, right, otuse, or not tringle. ute right otuse 5 3 not tringle 6 ute right otuse not tringle

Slide 37 / 240 13 Tell whether the lengths 35, 65, nd 56 represent the sides of n ute, right, or

Slide 38 / 240 14 Tell whether the lengths represent the sides of n ute, right, or ute right

then tringle is right. Similrity in Right Tringles If 2 > 2 + 2, then tringle is otuse.

Return to the Tle of ontents Slide 41 / 240 There re mny proofs to the Pythgoren Theorem.

7 Slide 37 / Tell whether the lengths 35, 65, nd 56 represent the sides of n ute, right, or otuse tringle. Slide 38 / Tell whether the lengths represent the sides of n ute, right, or otuse tringle. ute right otuse ute tringle right tringle otuse tringle Slide 39 / 240 Slide 40 / 240 Review If 2 = 2 + 2, then tringle is right. Similrity in Right Tringles If 2 > 2 + 2, then tringle is otuse. If 2 < 2 + 2, then tringle is ute. Return to the Tle of ontents Slide 41 / 240 There re mny proofs to the Pythgoren Theorem. How mny do you know? Slide 42 / 240 Theorem The ltitude of right tringle divides the tringle into two smller tringles tht re similr to the originl tringle nd eh other. Tringle similrity n e used to prove the Pythgoren Theorem. How? is the ltitude of ~ ~

Given: is right tringle is the ltitude of Sttements is right tringle is right ngle Resons lik Given lik Given ef likof ltitude Therefore, the ltitude of right tringle divides the tringle into two

8 Slide 43 / 240 To prove this, lik for L 1 - Similr Right Tringles Let's prove the Theorem. Slide 44 / 240 The ltitude of right tringle divides the tringle into two smller tringles tht re similr to the originl tringle nd eh other. Given: is right tringle is the ltitude of Sttements is right tringle is right ngle Resons lik Given lik Given ef likof ltitude Therefore, the ltitude of right tringle divides the tringle into two smller tringles tht re similr lik to the originl tringle nd similr lik to eh other. Prove: ~ ~ is right ngle ~ is right ngle ~ ~ ~ ef of Perp Lines. 2 lines tht form rt lik ngle lik ll rt ngles re lik Refleive Prop of lik ~ lik ef of Perp Lines lik ll rt ngles re lik Refleive Prop of lik ~ lik Trnsitive Prop of ~ Slide 45 / 240 Slide 46 / 240 Let's sketh the 3 tringle's seprtely, with the sme orienttion. Mth up the ngles. d e ssign lengths to ll the segments. Let the lengths of the segments on the hypotenuse e d nd e. e d Lel the sides of tringle with the lower se letter of the opposite ngle. ~ ~ Helpful tip: If you set, then you n ssign ll the ngles vlue nd esily find the mthes. euse the tringles re similr the orresponding sides re proportionl. ~ ~ Slide 47 / 240 Slide 48 / 240 To prove the Pythgoren Theorem, use the proportions. Given: Prove: is right tringle. is n ltitude. ~ Sttements Resons ltitude of rt tringle theorem. lik efinition of similr tringles. lik Using the multiplition property of equlity, multiply lik the eqution y. To prove the Pythgoren Theorem, use the proportions (ontinued). Given: Prove: is right tringle. is n ltitude. Sttements Resons Using the ddition property of equlity, dd eqution (1) lik nd eqution (2) together. lik istriutive Property lik Given lik Sustitution d e ~ (1) simplify lik ltitude of rt tringle theorem. lik efinition of similr tringles. lik Using the multiplition property of equlity, lik multiply the eqution y d e lik Simplify (2) lik simplify

Slide 49 / 240 Emple Find the length of the ltitude KI?

proportionl. K I J 5 K 13 = 60 4.62 Slide 51 / 240 Slide 52 / 240 Try this.

I S R R S H K J 5 P 3 4 Q S 4 Q P 5 S Slide 53 / 240 Slide 54 / 240 16 Find KJ.

Usully when we sk to find the men, we re sking for the rithmeti men.

9 Slide 49 / 240 Emple Find the length of the ltitude KI? H H Slide 50 / 240 It mye helpful to sketh the 3 tringle's seprtely, with the sme orienttion. I 13 5 K H 13 H K 5 I 13 J K 5 J J I euse the tringles re similr the orresponding sides re proportionl. K I J 5 K 13 = Slide 51 / 240 Slide 52 / 240 Try this... Find the length of RS. P 3 Q R Whih rtio is the rtio of orresponding sides? I S R R S H K J 5 P 3 4 Q S 4 Q P 5 S Slide 53 / 240 Slide 54 / Find KJ. The net two theorems re Geometri Men Theorems. I 7 24 Wht is men? n verge. Usully when we sk to find the men, we re sking for the rithmeti men. H K J Wht is n rithmeti men? The sum of n vlues divided y the numer of vlues (n). 25 Wht is geometri men? The nth root of produt of n vlues. It is defined for only positive numers (no negtive numers, no zero) For more informtion lik on this link: rithmeti Men vs Geometri Men

Slide 55 / 240 Slide 56 / 240 The geometri men of two positive numers nd is the positive numer tht stisfies = 2 = = Emple Find

2 = 8(14) Visully, the geometri men nswers this question: given retngle with sides nd, find the side of the squre whose re

Write the nswer is simplest rdil form. Slide 58 / 240 18 Find the geometri men of 3 nd 48.

10 Slide 55 / 240 Slide 56 / 240 The geometri men of two positive numers nd is the positive numer tht stisfies = 2 = = Emple Find the geometri men of 8 nd = 8(14) Visully, the geometri men nswers this question: given retngle with sides nd, find the side of the squre whose re equls tht of the retngle. 2 = 1 (only the positive vlue) Slide 57 / Find the geometri men of 7 nd 56. Write the nswer is simplest rdil form. Slide 58 / Find the geometri men of 3 nd 48. Students type their nswers here Slide 59 / 240 Slide 60 / 240 orollry The ltitude drwn to the hypotenuse of right tringle divides the the hypotenuse into two segments. The ltitude is the geometri men of the two segments formed. Emple Find z. 8 6 z is the ltitude of Sine, ~ 2 = ()

11 Slide 61 / 240 Slide 62 / 240 Emple Find z. 18 z 6 Try this... Find y. 1) 18 8 y 2) y 9 Slide 63 / 240 Slide 64 / Find. 20 Find Slide 65 / 240 orollry If the ltitude drwn to the hypotenuse of right tringle, divides the hypotenuse into two segments. The length of eh leg of the originl tringle is the geometri men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djent to the leg. Emple Find. Slide 66 / 240 R 4 S 9 T U is the ltitude of Sine, ~ ~ ~ ~ = =

12 Slide 67 / 240 Slide 68 / 240 Emple Find. E 6 4 F 21 Is PR geometri men etween QR nd SR? True P G Flse Q S R Slide 69 / Is the geometri men orret? True P Slide 70 / Whih proportion is orret? J K L Flse Q M S R Slide 71 / 240 Slide 72 / Find y. 25 Find y. 20 y y None of the ove 9

13 Slide 73 / 240 Slide 74 / Find. 5 8 Speil Right Tringles Return to the Tle of ontents Slide 75 / 240 In this setion you will lern out the properties of the two speil right tringles. 45 o o 30 o 90 o 45 o 60 o Slide 76 / tringle is n isoseles right tringle, where the hypotenuse is 2 times the length of the leg. hypotenuse = leg( 2) n you prove this? Tringle Theorem 45 o 2 45 o Slide 77 / 240 Slide 78 / 240 Emple Find the length of the missing sides. Write the nswer in simplest rdil form. P 45 o 6 Q Emple Find the length of the missing sides of the right tringle. S y T y o R V

14 Slide 79 / 240 Slide 80 / 240 Try this... Find the length of the missing sides. 27 Find the vlue of. 5 y 8 5# 2 (5# 2)/2 y 5 28 Find the vlue of y. 5 Slide 81 / 240 Slide 82 / Wht is the length of the hypotenuse of n isoseles right tringle, if the length of the legs is 8# 2 inhes. 5# 2 (5# 2)/2 y 5 Slide 83 / Wht is the length of eh leg of n isoseles, if the length of the hypotenuse is 20 m. In right tringle, the hypotenuse is twie the length of the shorter leg nd the longer leg is 3 times the length of the shorter leg. Slide 84 / Tringle Theorem 60 o # o hypotenuse = 2(shorter leg) longer leg = 3(shorter leg)

15 Slide 85 / 240 Slide 86 / 240 This n e proved using n equilterl tringle. For right tringle, is perpendiulr isetor. let =, = 2 nd = 60 o 2 # 3 30 o Emple Find the length of the missing sides of the right tringle. G 30 o = y = H 5 60 o F Slide 87 / 240 Slide 88 / 240 Rell tringle inequlity, the shortest side is opposite the smllest ngle nd the longest side is opposite the lrgest ngle. G 30 o Emple Find the length of the missing sides of the right tringle. M 60 o HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF y 9 y H 5 60 o F 30 o T Slide 89 / 240 Slide 90 / 240 Emple Find the re of the tringle. The ltitude (or height) divides the tringle into two 30 o -60 o -90 o tringles. h 14 ft 14 ft The length of the shorter leg is 7 ft. The length of the longer leg is 7 3 ft.?? = (h) = 14(7 # 3) # squre ft

16 Slide 91 / 240 Slide 92 / 240 Try this... Find the length of the missing sides of the right tringle. 15 Try this... Find the re of the tringle. 9 ft 30 o y 60 o 30 o Slide 93 / 240 Slide 94 / Find the vlue of. 32 Find the vlue of # 3 (7# 2)/ o 30 o 7# 3 (7# 2)/2 7# Slide 95 / 240 Slide 96 / Find the vlue of. 7 7# 3 (7# 2)/2 30 o o 34 The hypotenuse of 30 o -60 o -90 o tringle is 13 m. Wht is the length of the shorter leg? 14

Slide 97 / 240 35 The length the longer leg of 30 o -60 o -90 o tringle is 7 m. Wht is the length of the hypotenuse? Slide 98 / 240 Rel World Emple The wheelhir rmp t your shool hs height of 2.

30 o? 2.5 36 skteorder onstruts rmp using plywood. The length of the plywood is 3 feet long nd flls t n ngle of 45. Wht is the height of the rmp? Round to the nerest hundredth.

17 Slide 97 / The length the longer leg of 30 o -60 o -90 o tringle is 7 m. Wht is the length of the hypotenuse? Slide 98 / 240 Rel World Emple The wheelhir rmp t your shool hs height of 2.5 feet nd rises t ngle of 30 o. Wht is the length of the rmp? Slide 99 / 240 Slide 100 / 240 The tringle formed y the rmp is 30 o -60 o -90 o right tringle. The length of the rmp is the hypotenuse. 30 o? skteorder onstruts rmp using plywood. The length of the plywood is 3 feet long nd flls t n ngle of 45. Wht is the height of the rmp? Round to the nerest hundredth. hypotenuse = 2(shorter leg) hypotenuse = 2(2.5) hypotenuse = 5 The rmp is 5 feet long.? 45 o 3 feet Slide 101 / Wht is the length of the se of the rmp? Round to the nerest hundredth. Slide 102 / The yield sign is shped like n equilterl tringle. Find the length of the ltitude. 20 inhes 45 o 3 feet?

18 Slide 103 / 240 Slide 104 / The yield sign is shped like n equilterl tringle. Find the re of the sign. 20 inhes Trigonometri Rtios Return to the Tle of ontents Slide 105 / 240 Slide 106 / 240 Right tringle trigonometry is the study of the reltionships etween the sides nd ngles of right tringles. Ever sine the onstrution of the ell Tower in the 1100's, it hs slowly tilted south nd is t risk of flling over. If the ngle of slnt ever fll's elow 83 degrees, it is fered the tower will ollpse. Lening Tower of Pis, ell Tower in Pis, Itly Slide 107 / 240 Slide 108 / 240 Engineers n mesure the ngle of slnt using ny of the right tringles onstruted elow. Let's lulte the rtio's of the height to the se for eh right tringle. Engineers very refully mesure the perpendiulr distne from tower window (points, or F) to the ground (points G, E or ). Then they mesure the distne from the tower to points, E or G. Tringle Height se Rtio Height / se =50m =5m 50/5=10 E E=30m E=3m 30/3=10 FG FG=20m G=2m 20/2=10 nswer ~ E~ FG WHY? F Notie tht ll of the rtios re the sme. WHY? GE ngle of slnt The rtio of height/se is lso lled the slope rtio (rise/run) or tngent rtio.

Slide 109 / 240 Slide 110 / 240 When the tringle is dilted (pull sle), how does the ngle hnge? Wht hppens to the slope rtio? Wht hppens to the rtio when the ngle inreses?

19 Slide 109 / 240 Slide 110 / 240 When the tringle is dilted (pull sle), how does the ngle hnge? Wht hppens to the slope rtio? Wht hppens to the rtio when the ngle inreses? Wht hppens to the rtio when the ngle dereses? To lern right tringle trigonometry, first you need to e le to identify the sides of right tringle. Lel the sides of tringle with the lower se letter of the opposite ngle. lik for intertive wesite to investigte. In right tringle, there re 2 ute ngles. In the tringle to the left, nd re the ute ngles. Slide 111 / 240 Slide 1 / 240 Let's look t, when is the referene ngle, the side opposite is. the side djent (or net to) is. nd the hypotenuse is. dj opp hyp 40 Wht is the side opposite to J? JL J LK KJ L opp dj hyp When is the referene ngle, the side opposite is. the side djent (or net to) is. nd the hypotenuse is. K Slide 113 / 240 Slide 114 / Wht is the hypotenuse of the tringle? 42 Wht is the side djent to J? JL J L JL J L LK LK KJ KJ K K

Slide 115 / 240 Slide 116 / 240 43 Wht is the side opposite K? J L 44 Wht is the side djent to K?

20 Slide 115 / 240 Slide 116 / Wht is the side opposite K? J L 44 Wht is the side djent to K? J L JL JL LK KJ K LK KJ K Slide 117 / 240 Slide 118 / 240 Trigonometri Rtios trigonometri rtio is the rtio of the two sides of right tringle. There re 3 rtios for eh ute ngle of right tringle. The rtios re lled sine, osine, nd tngent (revited sin, os, nd tn). sinθ = The 3 Trigonometri Rtios opposite side hypotenuse This spells... SOHHTO osθ = or djent side hypotenuse tnθ = opposite side djent side θ whih is pneumoni to help you rememer the sides of right tringle (you'll need to rememer the spelling). Slide 119 / 240 Slide 0 / 240 lik for SOHHTO song on youtue.om "Gettin' Triggy Wit It". Emple Find the sin F, os F, nd tn F E 8 F Sine F is your referene ngle, lel the sides of the tringle opposite, djent nd hypotenuse. Use the pneumoni to find the trig rtios. opp 6 E dj 8 10 hyp F sinf = opp 6 hyp = 10 = 3 5 osf = dj = 8 4 = hyp 10 5 tn F = opp 6 3 = = dj 8 4 lwys redue frtions to lowest terms.

Slide 1 / 240 Slide 2 / 240 Emple Find the sin, os, nd tn. 6 10 45 Wht is the sin R? E 8 F 20/29 Sine is your referene ngle, lel the sides of the tringle opposite, djent nd hypotenuse.

21 Slide 1 / 240 Slide 2 / 240 Emple Find the sin, os, nd tn Wht is the sin R? E 8 F 20/29 Sine is your referene ngle, lel the sides of the tringle opposite, djent nd hypotenuse. Use the pneumoni to find the trig rtios. dj 6 E opp 8 10 hyp F sin = opp 8 4 = hyp 10 = 5 os = dj = 6 3 = hyp 10 5 tn = opp dj = 8 6 lwys redue frtions to lowest terms. 4 = 3 21/20 21/29 20/21 46 Wht is the osr? 20/29 21/20 21/29 20/21 Slide 3 / Wht is the tnr? 20/21 21/20 20/29 21/29 Slide 4 / Wht is the sinq? Slide 5 / Wht is the osq? Slide 6 / /29 21/20 21/29 29/20 20/29 21/20 21/29 29/21

3 To find the trigonometri rtio of n ngle, use lultor or trig tle.

deiml ples). Find the following: sin 84.3 =.9951 lik os 84.3 =.0993 lik tn 84.3 = 10.

Round to the nerest ten thousndth. 0.5 0.8660 1.7321 0.5774 52 Evlute os 60.

22 50 Wht is the tnq? 20/29 21/20 21/29 20/21 Slide 7 / 240 Slide 8 / 240 The ngle of slnt of the Tower of Pis is 84.3 To find the trigonometri rtio of n ngle, use lultor or trig tle. hek tht your lultor is set for degrees (not rdins) nd round your nswer to the ten thousndth ple (4 deiml ples). Find the following: sin 84.3 =.9951 lik os 84.3 =.0993 lik tn 84.3 = lik F ngle of slnt Slide 9 / 240 Slide 130 / Evlute sin 60. Round to the nerest ten thousndth Evlute os 60. Round to the nerest ten thousndth Slide 131 / Evlute tn 60. Round to the nerest ten thousndth Slide 132 / 240 Trig tles were used y erly mthemtiins nd stronomers to lulte distnes tht they were unle to mesure diretly. Tody, lultors re usully used

Slide 133 / 240 How do you find n unknown side mesure in right tringle when you re given n ute ngle nd one side? You need to identify the orret trig funtion tht will find the missing side.

23 Slide 133 / 240 How do you find n unknown side mesure in right tringle when you re given n ute ngle nd one side? You need to identify the orret trig funtion tht will find the missing side. Use SOHHTO to help. is your ngle of referene. Lel the given nd unknown sides of your tringle opp, dj, or hyp. Identify the trig funtion tht uses, the unknown side nd the given side. opp 30 o Slide 134 / 240 Emple Find the trig eqution tht will find. Using, I m looking for o nd I hve, so the rtio is o/ whih is tngent. now you n solve for, the missing side. dj Slide 135 / 240 Emple Find the trig eqution tht will find. Slide 136 / 240 Emple Find the trig eqution tht will find. 30 o 30 o Slide 137 / Using, whih is the orret trig eqution needed to solve for. sin40 = / os40 = / tn40 = / sin40 = / E 40 o Slide 138 / Using, whih is the orret trig eqution needed to solve for. sin50 = / os50 = / tn50 = / sin50 = / E 50 o

Slide 139 / 240 Slide 140 / 240 56 Using J, whih is the orret trig eqution needed to solve for.

24 Slide 139 / 240 Slide 140 / Using J, whih is the orret trig eqution needed to solve for. tn32 = /11 os32 = /11 tn32 = 11/ sin32 = 11/ J 32 o K 11 L 57 Using K, whih is the orret trig eqution needed to solve for. tn58 = /11 os58 = /11 tn58 = 11/ sin 58 = 11/ J 58 o K 11 L Slide 141 / 240 Slide 142 / 240 Finding the Missing Side of Right Tringle Now, you n solve for, the missing side. Round your nswer to the nerest tenth. Using your lultor, find the tn 84.3 Round your nswer to 4 deiml ples. You n rewrite with denomintor of 1 nd use the ross produt property or multiply oth sides of the eqution y 5 using the multiplition property of equlity (see net slide). opp Finding the Missing Side of Right Tringle Now, you n solve for, the missing side. Round your nswer to the nerest tenth. Multiply oth sides of the eqution y 5 using the multiplition property of equlity. opp dj dj Slide 143 / 240 Emple Find. Round your nswer to the nerest hundredth. Slide 144 / 240 Emple Find. Round your nswer to the nerest hundredth. G 25 o E G E M 65 o M

25 Slide 145 / 240 Emple Find y. Round your nswer to the nerest hundredth. Slide 146 / Find the length of LM. Round your nswer to the nerest tenth. P o E y L 68 o M Slide 147 / Find the length of LP. Round your nswer to the nerest tenth. P Slide 148 / 240 Eplin nd use the reltionship etween the sine nd osine of omplementry ngles. L 68 o M Slide 149 / 240 Slide 150 / 240 Find the mesure of?

Slide 151 / 240 60 For right tringle, wht is the mesure of? 30 degrees 50 degrees 60 degrees nnot e determined 30 o Slide 152 / 240 61 If the, find the omplementry ngle?

26 Slide 151 / For right tringle, wht is the mesure of? 30 degrees 50 degrees 60 degrees nnot e determined 30 o Slide 152 / If the, find the omplementry ngle? 20 degrees 70 degrees 160 degrees none of the ove Slide 153 / 240 Let's ompre the sine nd osine of the ute ngles of right tringle. In right tringle, the ute ngles re omplementry. m + m = = 90 sin = 4/5 sin 53.1 =.7997 os = 4/5 os 36.9 =.7997 sin = os 53.1 sin 53.1 = os The sine of n ngle is equl to the osine of its omplement. os = 3/5 os 53.1 =.6004 sin = 3/5 sin 36.9 =.6004 os = sin os 53.1 = sin 36.9 The osine of n ngle is equl to the sine of its omplement Slide 154 / 240 First, find the mesure of LP using the sine funtion. Then, find the mesure of LP using the osine funtion. sine funtion osine funtion L 68 o Sine nd osine re lled o-funtions of eh other. o-funtions of omplementry ngles re equl. 22 o P M Slide 155 / Given tht sin 10 =.1736, write the osine of omplementry ngle. sin 10 =.1736 sin 80 =.9848 os 10 =.9848 os 80 =.1736 Slide 156 / Given tht os 50 =.6428, write the sine of omplementry ngle. sin 50 =.7660 sin 40 =.6428 os 50 =.6428 os 40 =.7660

27 Slide 157 / Given tht os 65 =.4226, write the sine of omplementry ngle. Slide 158 / Wht n you onlude out the sine nd osine of 45 degrees? Students type their nswers here sin 25 =.4226 os 25 =.9063 sin 65 =.9063 os 65 =.4226 Slide 159 / 240 Slide 160 / 240 To solve right tringle mens to find ll 6 vlues in tringle. The lengths of ll 3 sides nd the mesures of ll 3 ngles. Solving Right Tringles Return to the Tle of ontents Slide 161 / 240 Slide 162 / 240 Let's solve right tringle given the length of one side nd the mesure of one ute n gle (S). You need to find the 3 missing prts. First, let's find the mesure of. 15 y 15 y 64 o z 64 o z

Slide 163 / 240 Slide 164 / 240 Then, let's find the mesure of. Then, let's find the mesure of. 15 26 y 15 26 13.48 64 o z 64 o z Slide 165 / 240 Try this... Find the missing prts of the tringle.

ut, how do you find the mesure of nd? 9 You will need to use the inverse trig funtions. If sinθ =, θ = sin -1 If osθ =, θ = os -1 Pronouned inverse sine, inverse osine, nd inverse tngent.

28 Slide 163 / 240 Slide 164 / 240 Then, let's find the mesure of. Then, let's find the mesure of y o z 64 o z Slide 165 / 240 Try this... Find the missing prts of the tringle. Slide 166 / 240 Let's solve right tringle given the length of two sides (SS). 11 R 9 E 37 o 15 y z Slide 167 / 240 Slide 168 / 240 First, find the length of sine we know how to do tht. ut, how do you find the mesure of nd? 9 You will need to use the inverse trig funtions. If sinθ =, θ = sin -1 If osθ =, θ = os -1 Pronouned inverse sine, inverse osine, nd inverse tngent. 15 y z If tnθ =, θ = tn -1 With the sine, osine nd tngent trig funtions, if you know the ngle θ nd the mesure of one leg, then you n find the mesure of leg of tringle. With the inverse sine, inverse osine nd inverse tngent trig funtions, if you know the mesures of 2 legs of tringle, you n find the mesure of the ngle. θ

θ = sin -1 ( opposite side ) hypotenuse Slide 169 / 240 The 3 Inverse Trigonometri Rtios θ = os -1 ( djent side ) hypotenuse θ = tn -1 ( opposite side ) djent side Slide 170 / 240 66 Find sin -1 0.8.

29 θ = sin -1 ( opposite side ) hypotenuse Slide 169 / 240 The 3 Inverse Trigonometri Rtios θ = os -1 ( djent side ) hypotenuse θ = tn -1 ( opposite side ) djent side Slide 170 / Find sin Round the ngle mesure to the nerest hundredth. Use the inverse trig funtion to find the unknown ngle mesure when you know the length of 2 sides. Rememer: θ Slide 171 / Find tn Round the ngle mesure to the nerest hundredth. Slide 172 / Find os Round the ngle mesure to the nerest hundredth. Slide 173 / 240 To find n unknown ngle mesure in right tringle, You need to identify the orret trig funtion tht will find the missing vlue. Use SOHHTO to help. is your ngle of referene. Lel the two given sides of your tringle opp, dj, or hyp. Identify the trig funtion tht uses, nd the two sides. Using osine. θ 9 15 hyp, I hve nd h, so the rtio is /h whih is now you n solve for, the missing ngle using the inverse trig funtion. dj Slide 174 / 240 Emple Find the trig eqution tht will find θ. 7 θ How re you going to lulte the mesure of?

30 θ 10 Slide 175 / 240 Emple Find the trig eqution tht will find θ. θ Slide 176 / 240 Emple Find the trig eqution tht will find θ. 9 Slide 177 / Whih is the orret trig eqution to solve for Slide 178 / Whih is the orret trig eqution to solve for 7 5 E E Slide 179 / 240 Slide 180 / Whih is the orret trig eqution to solve for Try this... Solve the right tringle. Round your nswers to the nerest hundredth. R K 11 Q 24 7 S J 9 L

31 Slide 181 / 240 Slide 182 / Find E. 73 Find m. 5 E 5 E 8 8 Slide 183 / 240 Slide 184 / Find the m E. 75 Find the m G. 5 E L 20 o 8 18 G Slide 185 / 240 Slide 186 / Find L. L 77 Find the m P o P 20 o o o o E 18 N 18 G

32 Slide 187 / 240 Slide 188 / Find RT S 8 40 o R T ngle of Elevtion nd epression Return to the Tle of ontents Slide 189 / 240 Slide 190 / 240 How n you use trigonometri rtios to solve word prolems involving ngles of elevtion nd depression? When you look up t n ojet, the ngle your line of sight mkes with line drwn horizontlly is the ngle of elevtion. Slide 191 / 240 Slide 192 / 240 The ngle of elevtion nd the ngle of depression re oth mesured reltive to prllel horizontl lines, they re equl in meure. When you look down t n ojet, the ngle your line of sight mkes with line drwn horizontlly is the ngle of depression.

Slide 193 / 240 Slide 194 / 240 79 How n you desrie the ngle reltionship etween the ngle of elevtion nd the ngle of depression?

The kite's string is 158 feet long nd my's rm is 3 feet off the ground. How high is the kite off the ground?

8480 = 158 158 158 Emple You re stnding on mountin tht is 5306 feet high. You look down t your mpsite t ngle of 30 o.

134 + 3 = 137 30 o 6 ft The kite is out 137 feet off the ground. 5306 ft Slide 197 / 240 Slide 198 / 240 Try this... 53 ft tn30 =.

33 Slide 193 / 240 Slide 194 / How n you desrie the ngle reltionship etween the ngle of elevtion nd the ngle of depression? orresponding ngles lternte interior ngles lternte eterior ngles none of the ove Emple my is flying kite t n ngle of 58 o. The kite's string is 158 feet long nd my's rm is 3 feet off the ground. How high is the kite off the ground? 158 feet 58 o 3 feet Slide 195 / 240 Slide 196 / ft 58 o sin# = sin58 =.8480 = Emple You re stnding on mountin tht is 5306 feet high. You look down t your mpsite t ngle of 30 o. If you re 6 feet tll, how fr is the se of the mountin from the mpsite? = 134 Now, we must dd in my's rm height = o 6 ft The kite is out 137 feet off the ground ft Slide 197 / 240 Slide 198 / 240 Try this ft tn30 =.5774 = You re looking t the top of tree. The ngle of elevtion is 55 o. The distne from the top of the tree to your position (line of sight) is 84 feet. If you re 5.5 feet tll, how fr re you from the se of the tree? 30 o.5774 = 53 9,200 ft The mpsite is out 9,200 ft from the se of the mountin.

34 Slide 199 / When you look down t n ojet, the ngle your line of sight mkes with line drwn horizontlly is the ngle of. elevtion depression Slide 200 / Ktherine looks down out of the rown of the sttue of lierty to n inoming ferry out 345 feet. The distne from rown to the ground is out 250 feet. Wht is the ngle of depression? Slide 201 / 240 Slide 202 / Wht is the distne from the ferry to the se of the sttue? Lw of Sines nd Lw of osines Return to the Tle of ontents Slide 203 / 240 How n you solve non-right tringle? How n you find the side lengths nd ngle mesures of non-right tringles? The Lw of Sines nd Lw of osines n e used to solve ny tringle. You n use the Lw of Sines when you re given - 1. Two ngle mesures nd ny side length (S or S) 2. Two side lengths nd the mesure of non-inluded ngle (SS) when the ngle is right ngle or n otuse ngle. The Lw of Sines hs prolem deling with SS when the ngle is ute. There n e zero, one or two solutions. lik on: Khn demy Video "More On Why SS Is Not Postulte" for more info. You n use the Lw of osines when you re given - 3. Three side lengths (SSS) 4. Two side lengths nd the mesure of n inluded ngle (SS) Lw of Sines Slide 204 / 240 If hs sides of length,, nd, then sin = sin = sin To use the Lw of Sines, 2 ngles nd 1 side must e given.

Let's prove the Lw of Sines If hs sides of length,, nd, then sin = sin = sin Given: hs sides of length,, nd Prove: sin = sin = sin Slide 205 / 240 h Sttements with side lengths,, nd rw n ltitude from

Sttements rw n ltitude from to side h Let g e the length of the ltitude g Resons lik ef of ltitude lik ef of sine Multiply lik y. Mult Prop of =. Multiply lik y. Mult Prop of =. Sustitution lik Prop of = ivide y.

35 Let's prove the Lw of Sines If hs sides of length,, nd, then sin = sin = sin Given: hs sides of length,, nd Prove: sin = sin = sin Slide 205 / 240 h Sttements with side lengths,, nd rw n ltitude from to side Let h e the length of the ltitude Resons lik Given ef of ltitude lik lik ef of sine Prove the Lw of Sines (ontinued) Slide 206 / 240 Given: hs sides of length,, nd Prove: sin = sin = sin Sttements rw n ltitude from to side h Let g e the length of the ltitude g Resons lik ef of ltitude lik ef of sine Multiply lik y. Mult Prop of =. Multiply lik y. Mult Prop of =. Sustitution lik Prop of = ivide y. lik ivision Prop of = Multiply lik y. Mult Prop of =. Multiply lik y. Mult Prop of =. Sustitution lik Prop of = ivide y. ivision Prop of = lik Sustitution lik Prop of = Slide 207 / 240 Slide 208 / 240 Use the Lw of Sines to solve the tringle. 65 o 70 o 10 sin = sin = sin Selet the rtios sed on the given informtion. Given: m, m nd (side ) (S) Whih rtios must e used first? Why? First we n find the length side. sin = sin sin70 = sin o 70 o 10 Slide 209 / 240 Slide 210 / 240 efore we find the length of side, we find the m. 70 o 10 Now we find the length side. 70 o o Tringle Sum Theorem m + m + m = 180 o =10.37 sin = sin 65 o =45 o =10.37

36 Slide 211 / 240 Try this... Use the Lw of Sines to find the length of side (S). Slide 2 / 240 Emple... Find the length of side (SS with n otuse ngle). Sine the length of the side opposite < is given, hint find the m< first o o 29 o 9 Slide 213 / 240 Slide 214 / Find the m. 19 o 31 o 29 o 28 o 70 o o 84 Whih rtio must e used to find the length of or? sin o 81 o sin sin sin 85 Wht is the length of? Slide 215 / Wht is the length of? Slide 216 / o o o 81 o

Lw of osines Slide 217 / 240 If hs sides of length,, nd, then: To use the Lw osines, you must e given the length of 3 sides (SSS) or the length of 2 sides nd the mesure of the inluded ngle (SS).

ltitude from to side. Let h e the length of the lt. Let e the length of. Then (-) is the length of.

37 Lw of osines Slide 217 / 240 If hs sides of length,, nd, then: To use the Lw osines, you must e given the length of 3 sides (SSS) or the length of 2 sides nd the mesure of the inluded ngle (SS). Let's prove the Lw of osines If hs sides of length,, nd, then Given: hs sides of length,, nd Prove: (similr resoning shows tht ) Slide 218 / 240 Sttements with side lengths,, nd Given lik rw n ltitude from to side. Let h e the length of the lt. Let e the length of. Then (-) is the length of. In, os = / (1) =(os) h Resons lik ef of ltitude - Segment lik ddition Postulte efinition lik of osine Multiply lik y. Mult Prop of =. (2) In, Pythgoren lik Theorem In, Pythgoren lik Theorem lik Simplify lik Sustitution, eqution (2) ssoitive lik Prop of ddition Sustitution, lik eqution (1) Slide 219 / 240 Slide 220 / 240 Emple Use the Lw of osines to solve the right tringle. =16 =27 =16 =27 =23 is opposite < is opposite < is opposite < The formul you hoose depends on whih ngle you re solving for first. To find the m, 2 = (os) 16 2 = (23)(27)(os) 256 = (os) 256 = 58-42(os) = -42(os).8068 = os = os -1 (.8068) m o =23 Slide 221 / 240 Slide 222 / 240 =16 = =16 =27 To find the m, 2 = (os) 23 2 = (16)(27)(os) 529 = (os) 579 = (os) -406 = -864(os).4699 = os =os -1 (.4699) m o =23 Using 2 different methods, the nswers re slightly different euse of rounding. or =23 To find the m, Use the Tringle Sum Theorem.

38 Slide 223 / 240 Try this... Use the Lw of osines to find the m< (SSS). Slide 224 / In the tringle the length of is Slide 225 / In the tringle the length of is... Slide 226 / Whih formul would you use to find the m<? 8 2 = (os) = (os) 2 = (os) 2 = (os) 90 Wht is the m? Slide 227 / Wht is the m? Slide 228 /

Slide 229 / 240 92 Wht is the mesure of (S)? Students type their nswers here 4 50 8 Slide 230 / 240 93 The Lw of Sines nd osines is used to solve.

39 Slide 229 / Wht is the mesure of (S)? Students type their nswers here Slide 230 / The Lw of Sines nd osines is used to solve... right tringles ute tringles otuse tringles ll tringles Slide 231 / 240 Slide 232 / 240 o you rememer this? Previously, we found the re of tringle when we were given 3 sides. Find the re of the tringle. re of n Olique Tringle 13 feet 13 feet Return to the Tle of ontents 10 feet 1 = 2 h Slide 233 / 240 is the se of the tringle = 10. h is the ltitude (or height). It is the perpendiulr isetor of the se in n isoseles tringle. Find h, using the pythgoren theorem - Slide 234 / 240 Wht formul n you use to find the re of tringle if you know the length of two sides nd the mesure of n inluded ngle (SS)? Find the re of the tringle. 13 feet 13 feet h 13 feet feet 5 feet 10 feet

Slide 235 / 240 Sine = 1 h nd = 10, we need to find h. 2 13 feet 67.38 h 10 feet Slide 236 / 240 Let's derive the formul for n olique tringle. Given: hs sides of length,, nd. ltitude h.

Formul for the lik re of tringle. Sustitution lik Prop of = ommuttive Prop of Multiplition lik Slide 237 / 240 Slide 238 / 240 95 Find the re of the tringle to the nerest tenth.

40 Slide 235 / 240 Sine = 1 h nd = 10, we need to find h feet h 10 feet Slide 236 / 240 Let's derive the formul for n olique tringle. Given: hs sides of length,, nd. ltitude h. Prove: Sttements Resons with side lengths,, nd Given lik h rw n ltitude from to side ef likof ltitude Let h e the length of the ltitude ef likof sine Multiply lik y. Mult Prop of =. efinition. Formul for the lik re of tringle. Sustitution lik Prop of = ommuttive Prop of Multiplition lik Slide 237 / 240 Slide 238 / Find the re of the tringle to the nerest tenth. Students type their nswers here Slide 239 / Find the re of the tringle to the nerest tenth. Students type their nswers here Slide 240 / Find the re of the tringle to the nerest tenth. Students type their nswers here

Geometry Proofs: Chapter 7, Sections 7.1/7.2

Geometry Proofs: Chapter 7, Sections 7.1/7.2 Pythgoren Theorem: Proof y Rerrngement of re Given: Right tringle with leg lengths nd, nd hypotenuse length. Prove: 2 2 2 = + Proof #1: We re given figures I nd II s ongruent right tringles III with leg