FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 1 MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) TRINGLE = 2-dimentionl she hving 3 sides nd 3 ngles. HRTERISTI OF TRINGLES I) Every tringle is n enclosed she tht hs these three chrcteristics: 1) three ngles 2) three sides 3) the sum of the lengths of the short sides is greter thn the length of the long side. ) onsider the third chrcteristic common to ll tringles: 3) the sum of the lengths of the two short sides is greter thn the length of the long side. In order for tringle to be n enclosed she, the sum of the length of the two short sides must be greter thn the length of the long side. If the sum of the length of the two short sides is less thn the length of the long side, then the two short sides re not long enough to connect to ech other nd the she is not tringle. If the sum of the length of the two short sides equl to the length of the long side, then when ll three sides re connected, the two short sides re not long enough to enclose sce nd is not tringle. 1) onsider Δ : 3 cm 5 cm 4 cm The sum of the lengths of the short sides is lrger thn the length of the long side: 3 cm + 4 cm >5 cm 7 cm >5 cm This mens the three sides enclose sce nd the she is tringle. 2) onsider the three lines hving these lengths: 1 cm, 2 cm, nd 5 cm. The sum of the lengths of the short sides is shorter thn the length of the long side: 1 cm 2 cm 1 cm + 2 cm >5 cm 5 cm 3 cm > 5 cm This mens the three sides do not enclose sce, thus the she is not tringle. ) SLE PROLEMS 1: Study these emles crefully. e sure you understnd nd memorize the rocess used to comlete them. INSTRUTIONS: Determine if ech grou of three lengths cretes tringle. Justify. 1. 5, 6, 8 2. 7, 3, 11 5+ 6 >8 11>8 3+ 7 >11 10 > 11 ecuse the sum of the short sides is bigger thn the long side, the three sides will form tringle. ecuse the sum of the short sides is not bigger thn the long side, the three sides will not form tringle. ) REQUIRED PRTIE 1: INSTRUTIONS: Determine if ech of these three lengths cretes tringle. Justify. {nswers re on ge 9 of these notes.} 1. 1, 2, 3 2. 20, 12, 16 3. 6, 9, 16 4. 8, 20, 3 II) NMING THE PRTS OF TRINGLE ) The ngles of tringle re nmed using citl letters. The sides of tringle cn be nmed using the citl letters tht nme the ngles t ech end of the side or by using the lower cse letter of citl letter nming the ngle tht is found oosite the side. Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 2 1) SLE PROLEM 2: Study this emle crefully. e sure you understnd nd memorize the rocess used to comlete it. INSTRUTIONS: Drw nd lbel comletely Δ. = b = c = c b 2) REQUIRED PRTIE 2: INSTRUTIONS: Drw nd lbel comletely these tringles. {nswers re on ge 9 of these notes.} 1. Δ DEF 2. Δ XYZ 3. Δ LRX 4. Δ T MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) RIGHT TRINGLE = tringle hving one 90 ngle. 2) OLIQUE TRINGLE = tringle tht lcks 90 ngle. 3) UTE TRINGLE = tringle where ll the ngles re less thn 90. 4) OTUSE TRINGLE = tringle hving one ngle lrger thn 90. TYPES OF TRINGLES I) LSSIFYING TRINGLES ) There re mny different tyes of tringles. Mthemticins clssify tringles into two brod ctegories nd ultimtely three tyes in order to determine how to solve them. Use this flow chrt to clssify tringles. TRINGLE HS 90 NGLE LKS 90 NGLE RIGHT TRINGLE OLIQUE LL NGLES LESS THN 90 ONE NGLE LRGER THN 90 UTE TRINGLE OTUSE TRINGLE 1) SLE PROLEMS 3: Study these emles crefully. e sure you understnd nd memorize the rocess used to comlete them. INSTRUTIONS: lssify these tringles s right tringle, oblique-cute tringle, or oblique-obtuse tringle. 1. nswer: This is right tringle becuse it hs 90 ngle. 90 ontinued on the net ge. Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 3 2. nswer: The hsh mrks indicte tht this is n Isosceles tringle, therefore = 25. Tringle sum theory gives = 130, thus this is n oblique-obtuse tringle. 25 2) REQUIRED PRTIE 3: INSTRUTIONS: lssify these tringles s right tringle, oblique-cute tringle, or oblique-obtuse tringle. {nswers re on ge 9 of these notes.} 1. Y 2. X 90 Z M G 3. 4. F 95 59 45 P D 45 H 79 MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) RIGHT TRINGLE = tringle hving one 90 ngle. 2) HYPOTENUSE = side of tringle directly oosite the 90 ngle. 3) LEGS OF TRINGLE = the sides of tringle directly oosite the cute ngles (ngles less thn 90 ); the sides of the tringle tht touch the 90 ngle. 4) PYTHGEN THEEM = formul tht is used to determine the unknown length of one side of right tringle when the lengths of the other two sides re known. leg 2 +leg 2 = hyotenuse 2 is commonly written s 2 +b 2 = c 2 I) Every right tringle hs these three chrcteristics: 1: one 90 ngle 2: two cute ngles 3: one hyotenuse, which is lwys the side tht is directly oosite the 90 ngle. ) The side of the right tringle tht is oosite the 90 ngle is the longest side nd is clled the hyotenuse. The sides of the right tringle tht touch the 90 ngle re clled legs. Study this right tringle. e sure you understnd nd memorize how to identify the right tringle s hyotenuse nd its legs. leg hyotenuse NOTE: The bo t indictes the ngle is 90. leg II) THE PYTHGEN THEEM ) The Pythgoren theorem is this formul: leg 2 +leg 2 = hyotenuse 2. The Pythgoren theorem is often written s: 2 +b 2 = c 2, where sides nd b re the right tringle s legs nd side c is the hyotenuse. The Pythgoren theorem is cn be used to determine if tringle is right tringle, nd to determine the length of side of right tringle when the lengths of ech of the other two sides re known. Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 4 1) SLE PROLEMS 4: Study these emles crefully. e sure you understnd nd memorize the rocess used to comlete them. 1. Is this right tringle? Use the Pythgoren theorem to determine if this is right tringle. 3 cm 5 cm leg 2 +leg 2 = hyotenuse 2 3 2 + 4 2 = 5 2 4 cm 9 + 16 = 25 25 = 25 ecuse the left side of the eqution is equl to the right side of the eqution, the sttement is true nd the tringle is right tringle. 2. tringle hs these side lengths: 23 mm, 49 mm nd 60 mm. Is it right tringle? Use the Pythgoren theorem to determine if this is right tringle. leg 2 +leg 2 = hyotenuse 2 23 2 + 49 2 = 60 2 529 + 2401 = 3600 2930 3600 ecuse the left side of the eqution does not equl to the right side of the eqution, the sttement is not true nd the tringle is not right tringle. 2) REQUIRED PRTIE 4: INSTRUTIONS: nswer these questions. Justify your nswers by including rorite clcultions. {nswers re on ge 9 of these notes.} 1. Is this right tringle? 2. Is this right tringle? 9 cm 21 cm 13 cm 8 mm 17 mm 15 mm 3. tringle hs these side lengths: 5 cm, 12 cm nd 13 cm. Is it right tringle? 4. tringle hs these side lengths: 15 km, 20 km nd 30 km. Is it right tringle? 3) SLE PROLEMS 5: Study these emles crefully. e sure you understnd nd memorize the rocess used to comlete them. INSTRUTIONS: Determine the length of the unknown side in ech of these right tringles. Round deciml nswers to one deciml lce. 1. 9 mm nswer: Use the Pythgoren theorem to determine length of side. leg 2 +leg 2 = hyotenuse 2 90 9 2 + 12 2 = 2 12 mm 81 + 144 = 2 225 = 2 225 = 2 15 = = 15 mm ontinued on the net ge. Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 5 2. nswer: Use the Pythgoren theorem to determine length of side. leg 2 +leg 2 = hyotenuse 2 34 m 42 m 2 + 34 2 = 42 2 2 + 1156 = 1764 2 +1156 1156 = 1764 1156 2 = 608 2 = 608 = 24.65765601...! 24.7 m 4) REQUIRED PRTIE 5: INSTRUTIONS: Find the length of the unknown side. Justify your nswers by including rorite clcultions. Round deciml nswers to one deciml lce. {nswers re on ge 9 of these notes.} 1. 2. 16 3. m 13 cm 29 mm 9 cm 23 m 43 mm MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) REFERENE NGLE = NGLE OF INTEREST = the right tringle s cute ngle, the mesure of which you re required to determine or must use to determine the required sine, cosine nd tngent rtio. 2) OPPOSITE LEG = the leg (side) of right tringle tht is directly oosite the reference ngle. 3) DJENT LEG = the leg (side) of right tringle tht is beside the reference ngle. The hyotenuse is never the djcent side. I) THE PRIMRY TRIGONOMETRI RTIOS: soh ch to ) The three rimry trigonometric rtios re clled sine, cosine nd tngent. They re shortened to sin, cos nd tn. MEMIZE THEM. sin, cos nd tn re frctions creted from the legs of right tringle reltive to the reference ngle lso clled the ngle of interest. The REFERENE NGLE is the right tringle s cute ngle, the mesure of which you re required to determine or must use to determine the required sine, cosine nd tngent rtio. Mthemticins often use the Greek citl letters thet θ, or bet β to identify the reference ngle. It is the reference ngle tht determines which is the oosite leg nd which is the djcent leg. refully study the tringles below. They re the sme right tringle with the reference ngle chnged. NOTIE how the nmes of the legs chnge s the reference ngle chnges from to. e sure you understnd nd memorize how to determine how to identify the oosite leg nd djcent leg reltive to the reference ngle correctly. Oosite leg θ djcent leg Reference ngle djcent leg Rob shby & Richrd Obyshi 2015. Duliction by ermission only. β Oosite leg
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 6 ) The three rimry trigonometric rtios for sine, cosine nd tngent re given here. 1) The sine of reference ngle thet, θ, is the frction/rtio creted by lcing the length of leg oosite ngle thet over the length of the hyotenuse. sine θ = sin θ = Oosite Hyotenuse sin θ = O H 2) The cosine of reference ngle thet, θ, is the frction/rtio creted by lcing the length of leg djcent to ngle thet over the length of the hyotenuse. cosine θ = length of djcent leg cos θ = djcent Hyotenuse cos θ = H 3) The tngent of reference ngle thet, θ, is the frction/rtio creted by lcing the length of leg oosite thet over the length of the leg djcent to ngle thet. tngent θ = lenght of djcent leg tn θ = Oosite djcent tn θ = O 4) The three rimry trigonometric rtios for sine, cosine nd tngent cn be memorized using this cronym: soh ch to, which mens sine equls oosite over hyotenuse - cosine equl djcent over hyotenuse - tngent equls oosite over djcent. IT IS STRONGLY REOMMENDED THT YOU MEMIZE THE RONYM SOH - H - TO Y WRITING IT T THE TOP OF EH PGE THT YOU RE OELTING TRIGONOMETRI LULTIONS. ) Using soh ch to 1) SLE PROLEM 6: Study these emles crefully. e sure you understnd nd memorize the rocess used to comlete them. INSTRUTIONS: nswer these questions. 1. Determine the sine, cosine nd tngent rtios/frctions for ngle thet, θ, nd for ngle bet, β. nswer: sine θ = c β b θ sin θ = c b = sine β = sin β = b = cosine θ = length of djcent leg cos θ = b = cosine β = length of djcent leg cosβ = c b = tngent θ = lenght of djcent leg tn θ = c = tngent β = lenght of djcent leg tn β = c = 2. Determine the sine, cosine nd tngent rtios/frctions for both cute ngles in this right tringle. M P m nswer: The cute ngles re M nd P. sine Μ = sin Μ = m = P sine P = sin P = = M cosine Μ = length of djcent leg cosμ = = M cosine P = length of djcent leg cos P = m = P tngent Μ = lenght of djcent leg tn Μ = m = P M tngent P = lenght of djcent leg tn P = m = M P Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 7 2) REQUIRED PRTIE 6: INSTRUTIONS: Determine the sine, cosine nd tngent rtios/frctions for cute ngles of these right tringles. {nswers re on ge 9 of these notes.} 1. M 2. P P m G i g I I) SOLVING RIGHT TRINGLES ) Solving right tringles requires the use of the Pythgoren theorem nd the rimry trigonometric rtios for sine, cosine nd tngent of one or both of the tringle s cute ngles (n ngle less thn 90 ) to find the unknown side length(s) nd the mesure of the unknown ngle(s). REMEMER: lwys use the informtion given in the question to determine the vlue of the informtion you need to clculte. Only use clculted vlues in future clcultions if bsolutely necessry. 1) USE THESE STEPS TO SOLVE RIGHT TRINGLES 1: List the ngles nd sides you know nd the ngles nd sides you don t know. 2: If you know two side lengths then find the third side length first. If you know the mesure of two of the ngles then find the mesure of the third ngle first. 3: Determine the vlues of the other missing informtion. 4: List the nswers - informtion you don t know. 2) SLE PROLEMS 7: INSTRUTIONS: Solve these right tringles to one deciml lce (the nerest tenth) 1. M Ste 1: KNOW DON T KNOW P m P = 90! M =?? = 42! = 16 cm =?? cm m =?? cm Ste 2: Find mesure of M =??. Ste 3: Find side lengths nd m. M = 180! 90! 42! M = 48! = 16 cm 42 Reltive to, = oosite leg, = hyotenuse, use soh. sin = Reltive to, m = djcent leg, = hyotenuse, use ch. cos = m sin 42! = 16 ( ) = 16 sin 42! ( ) = 16 sin 42! " 10.706 ( ) 16 16 cos 42! = m 16 16( cos 42! ) = m 16 ( ) m = 16 cos 42! m " 11.890 ( ) 16 Ste 4: List the nswers: M = 48!,! 10.7 cm, m! 11.9 cm Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 8 2. P Ste 1: KNOW DON T KNOW i = 11.2 m G P = 90! I =?? G = 54.3! i = 11.2 m ( ) g = 11.2 tn 54.3! g " 15.586 ( ) 11.2 11.2 g =?? m =?? m Ste 2: Find mesure of I =??. Ste 3: Find side lengths g nd. I = 180! 90! 54.3! Reltive to G, g = oosite leg, Reltive to G, = hyotenuse, i = djcent leg, use to. i = djcent leg, use coh. I = 35.7! tn G = g i cosg = i tn54.3! = g cos54.3! = 11.2 11.2 11.2( tn 54.3! ) = g ( cos54.3! ) = 11.2 ( ) Ste 4: List the nswers: I = 35.7!, g! 15.6 m,! 19.2 m 3. Ste 1: KNOW DON T KNOW c = 10 mm 54.3 = 90! =?? = 14 mm =?? c = 10 mm b =?? mm Ste 2: Find mesure of side length b. Ste 3: Find =?? nd =??. b 2 = 2 +c 2 b 2 = 14 2 + 10 2 b 2 = 196 + 100 b 2 = 296 b b 2 = 296 b! 17.204 g I = 14 mm Reltive to, = oosite leg, c = djcent leg, use to. tn = c tn = 14 10 = tn 1 ( 14 10 )! 54.462 " Ste 4: List the nswers: b! 17.2 mm,! 54.5!,! 35.5! ( cos54.3! ) = 11.2 cos 54.3! cos 54.3! = 11.2 cos54.3! " 19.193 Reltive to, c = oosite leg, = djcent leg, use to tn = c tn = 10 14 = tn 1 ( 10 14)! 35.537 " 3) REQUIRED PRTIE 7: INSTRUTIONS: Solve these right tringles. Record your nswers to the nerest tenth (first deciml lce). {nswers re on ge 9 of these notes.} 1. 2. E 3. 47 5.5 m 9 cm D 13 cm F 14.5 8.3 km Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 9 NSWERS TO THE REQUIRED PRTIE Required Prctice 1 from ge 1: 1) Not tringle 2) Is tringle 3) Not tringle 4) Not tringle Required Prctice 2 from ge 2 1) D 2) X 3) X 4) T f e z y l r E d Required Prctice 3 from ge 3 1) right tringle 2) right tringle 3) oblique-obtuse tringle 4) oblique-cute tringle Required Prctice 4 from ge 4 1) Not right tringle. 2) It is right tringle. 3) Is right tringle. 4) Not right tringle. Required Prctice 5 from ge 5 1)! 9.4 cm 2)! 28.0 m 3)! 31.7 cm F Y Z R L t c Required Prctice 6 from ge 7 1) sin Μ = m = P M, cosμ = = M, tn Μ = m = P, sin = = M, cos = m = M, tn = m = P 2) sin G = g = IP GI, cosg = i = GP GI, tn G = g i = GP IP, sin I = i = GP GI, cos I = g = IP GI, tn I = m = IP GP Prctice 7 from ge 8 1) = 43!, " 4.0 m, c " 3.8 m 2) e! 9.4 cm, E! 46.2 ", F! 43.8 " 3) = 75.5!, " 2.1 km, b " 8.6 km Rob shby & Richrd Obyshi 2015. Duliction by ermission only.
FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 10 SSIGNMENT: PRINT THIS INFMTION ON YOUR OWN GRID PPER LST then FIRST Nme T20 RIGHT TRINGE TRIGONOMETRY lock: Show the rocess required to comlete ech roblem to void receiving zero grde. Netness ounts!!! (Mrks indicted in itlicized brckets.) REMEMER TO USE GRID PPER F LL SSIGNMENTS!!! USE SLE OF 1 F LL GRIDS. Stte whether or not tringles cn be mde from these side length combintions. Justify. 1) 8, 9, 13 (2) 2) 18, 43, 25 (2) Solve these tringles. 3) 8 m 4) 55 P U 20 km 46 G (7.5) (8) Drw nd lbel then solve these tringles. 5) RED hs E = 90 ", d = 5 m nd e = 8 m (9.5) 6) TG hs = 90 ", g = 5 cm nd t = 12 cm (9.5) 7) P hs = 90 ", = 11 mm nd = 18 mm (9.5) /48 Rob shby & Richrd Obyshi 2015. Duliction by ermission only.