Lesson 18.2: Right Triangle Trigonometry

Transcription

1 Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to non-right tringles (hpter ), irles nd irulr motion, nd wide vriety of pplitions. s we shll see in the next unit, the six prts of ny tringle ( sides nd ngles) re inherently linked through the proesses of Trigonometry. Given tht the tringle is right tringle we know: one ngle is 90 ; the side the right ngle is the longest in the tringle (nd the smllest side is the smllest ngle); the remining two ngles re omplementry, nd so if we know one other ngle we know ll three ngles in the tringle; the sides re relted y Pythgors Theorem, nd so if we know ny two sides, we n find the third side of the tringle. Using Trigonometry, if we re given right tringle nd the length of ny two sides, we n determine the third side y using Pythgors Theorem. This in turn is suffiient informtion to lulte the trigonometri rtios of the ngles of the tringle nd onsequently the mesures of the ngles. 8..: THE RELTIONSHIP ETWEEN SIDES ND NGLES OF TRINGLE The sides of the tringle my lso e nmed ording to their reltionship to given ngle. In right tringle, the is lwys the side the right ngle. The side the ngle is the lled the side; while the side whih forms one rm of the ngle is lled the side. onsider the tringle shown. In reltion to the ngle, o the side is o the side is. DJENT OPPOSITE HYPOTENUSE In reltion to the ngle, o the side is OPPOSITE HYPOTENUSE o the side is. DJENT NOTE: These reltionships do not pply to the right ngle, hpter : Right Tringle Trigonometry Pge

2 onsider ny two right tringles. If the tringles hve one other equl ngle, x sy, then the third ngle of eh tringle, (90 - x), must lso e equl. Therefore the two tringles must e similr. y setion., sine the two tringles re similr we know tht pirs of orresponding sides re in fixed rtio. PR QR PR QR PQ PQ (90 - x ) P (90 - x ) Tking these rtios one t time: PR PQ o side to x QR PQ o side to x x R x Q PR QR side to x side to x o o Notie lso these rtios for n e relted to the ngle x or to the omplement of the ngle, (90 - x ): QR PQ o side to ( 90 x) nd PR PQ o side to ( 90 x) To mintin onsisteny, note tht PR QR gives us nd so QR PR QR side to ( 90 x) PR o side to ( 90 x) o euse these fixed rtios re very importnt to us we ssign them speil nmes, s you will see in the next setion. 8..: THE DEFINITIONS OF THE TRIGONOMETRI RTIOS IN RIGHT TRINGLE sin sin os os tn tn NOTE: Like ll rtios, the trigonometri rtios do not hve units. Whtever units re used to mesure the sides re nelled out during the division proess. hpter : Right Tringle Trigonometry Pge

3 Trigonometri Rtios of omplimentry ngles In right tringle the two non-right ngles must lwys sum to 90 nd thus re lwys omplimentry. In the right tringle, 90, the ngles nd re omplimentry, nd sin os nd os sin The side is to nd to ; the side is to nd to. Thus: side side sin os nd side os side sin Finding the Trigonometri Rtios in Right Tringle Exmple : In, 90, nd. Find: ) the length of the, nd ) the vlues of sin, os, tn. ) the vlues of sin, os, tn. First use Pythgors Theorem to find the length of the : For the, the side mesures units nd the side mesures units. DJENT OPPOSITE HYPOTENUSE sin os tn For the, the side mesures units nd the side mesures units. OPPOSITE DJENT HYPOTENUSE sin os tn NOTE: sin os nd os sin hpter : Right Tringle Trigonometry Pge

4 Exmple : In, 90, nd. Find () the length of the other leg (third side),nd () the vlues of sin, os, tn. () the vlues of sin, os, tn. First, drw digrm leling the dt given nd the unknown vlues. Use Pythgors Theorem to find the length of the third side of the tringle, NOTE : 0 For the, the side mesures nd the side mesures. HYPOTENUSE OPPOSITE sin os tn DJENT rtionlize the denomintor: tn For the, the side mesures nd the side mesures. HYPOTENUSE OPPOSITE DJENT sin os tn NOTE: sin os nd os sin hpter : Right Tringle Trigonometry Pge

5 Exmple : In PQR, R 90, p nd r. Find the length of the other leg (third side),of the tringle nd: () os Q () tn P Drw digrm leling the dt given nd the unknown vlues. Using Pythgors Theorem to find q R + q + q q q P q Q For Q, the side mesures unit; for P the side mesures units nd the side mesures unit. R osq tn P P Q Exmple : In XYZ, Z 90, x nd y 7. Find () the length of the z () tn Y () sin X Drw digrm leling the dt given nd the unknown vlues to e found. Using Pythgors Theorem to find z: + ( 7) z z z z Sustituting this vlue for z we n now find the trigonometri rtios: Y z Z 7 X tny 7 sin X Y 7 X Z hpter : Right Tringle Trigonometry Pge

6 Exmple : Using the lultor to pproximte vlues In, is right ngle,. nd.. Find () the length of the third side of the tringle, orret to one deiml ple. () sin, os, nd tn, pproximted orret to the nerest thousndth.. () sin, os, nd tn, pproximted orret to the nerest thousndth.. Using Pythgors Theorem to find : lultor entry: (. x. x ) orret to one deiml ple Sustituting this vlue for in we n find the six trigonometri rtios:..9 sin os tn sin os tn hpter : Right Tringle Trigonometry Pge

7 Exerise. In eh right tringle desried elow, 90. Find the ext vlues of sin, os nd tn.... In eh right tringle elow, find the ext vlue of sin, os, tn, nd sin, os, nd tn In, 90, 0, nd 0. Find the ext vlues of tn nd os 8. In PQR, R 90, p, nd r. Find the ext vlues of os P nd tn Q 9. In DEF, E 90, d.8, nd e.. Find the vlue of os D orret to three deiml ples. 0. In XYZ, X 90, x.8, nd y 7.. Find the vlue of tn Z orret to three deiml ples. hpter : Right Tringle Trigonometry Pge 7

8 8..: EXT TRIGONOMETRI RTIOS FOR 0, ND 0 NGLES. Ext Trigonometri Rtios for 0 nd 0. 0 Sine the sides of ny o 0 - o 0 - o 90 tringle re in the fixed rtio of : : we my use this tringle to determine the ext trigonometri rtios for 0 nd 0. From this tringle we see tht: 0 sin0 os 0 tn0 nd sin0 os 0 tn0 Ext Trigonometri Rtios for The sides of ny isoseles right tringle re in the fixed rtio of : :. From this tringle we see tht sin os tn Ext Trigonometri Rtios for 0 nd 90 0 With little it of imgintion we n oneptully derive the ext rtios for 0 nd 90. onsider right tringle, with nother ngle of 90. The third ngle must e 0. If this tringle hs one side of unit, the other side must e 0 units in length, nd the will e unit. From this tringle we see tht: sin0 0 os 0 0 tn0 0 nd sin90 0 os 90 0 tn90 undefined 0 Summry: sin 0 os 0 tn 0 undefined hpter : Right Tringle Trigonometry Pge 8

9 8..: USING THE LULTOR TO FIND TRIGONOMETRI RTIOS Most lultors work in one of two wys when omputing the vlues for the trigonometri funtions. You need to e fmilir with yours. First hek tht the MODE of the lultor is degrees [DEG] nd not rdins [RD] LULTOR TYPE LULTOR TYPE (older style). Press SIN, OS or TN. Enter numer in deiml degrees. Enter numer in deiml degrees. Press SIN, OS or TN. Press equl/enter to get the vlue Note: the equls sign is not needed here. Round the vlue to the required numer of deiml ples. Exmple : Find the vlue of, pproximted orret to the nerest thousndth: () sin 7 () os () tn.9 () Find sin 7 to the nerest thousndth Enter: lultor Type lultor Type sin 7 7 sin Disply: the numer of deiml ples given in the lultor s nswer will depend on your lultor:.998 Rounding: we wnt the nerest thousndth we pproximte orret to three deiml ples:.9 98 nd sine the digit in the fourth deiml ple is 9, we round the in the third ple up to. nswer: sin () Find os to the nerest thousndth Enter: lultor Type lultor Type os os Disply:.78 hpter : Right Tringle Trigonometry Pge 9

10 Rounding: pproximte orret to three deiml ples.7 8 sine the digit in the fourth deiml ple is, leve the numer in the third ple s. nswer: os 0.7 () Find tn.9 to the nerest thousndth Enter: lultor Type lultor Type tn.9.9 tn Disply:.7999 Rounding: pproximte orret to three deiml ples sine the digit in the fourth deiml ple is, round the in the third ple up to. nswer: tn. Exmple : Drw digrm of right tringle whih hs n ngle of. Find ) the size of the third ngle, nd ) the sine, osine nd tngent rtios of eh ngle orret to the nerest thousndth. 7 ) Sine the non-right ngles of right tringle re omplementry (i.e. dd to 90 ) the third ngle is: 90-7 (Or, sine the ngle sum of tringle is 80, the third ngle is: ) ) Using lultor: 0 sin os tn 0. sin 7 os 7 tn hpter : Right Tringle Trigonometry Pge 0

11 Exerise 8.. Questions 0: Use lultor to find eh of the following. pproximte ll nswers orret to the nerest thousndth (three deiml ples.) ) sin ) tn ) tn 87. ) os 7 ) os 8.8 ) sin 8 7) os.9 8) sin. 9) tn ) tn 8 Questions - : Use lultor to find the vlue of eh of following. pproximte ll nswers orret to the nerest thousndth (three deiml ples.) NOTE: In eh se, the trigonometri funtions re ofuntions of one nother, nd the ngles re omplementry ngles. ) sin, os 77 ) sin, os 8 ) sin 0, os 90 ) sin 90, os 0 Questions - 7: Drw digrm of right tringle whih hs the given ngle. Find ) the size of the third ngle, nd ) the sine, osine nd tngent rtios of eh ngle orret to the nerest thousndth. ) 0 ).7 7) Questions 8 0: omplete the following tle using the vlues omputed nd then use these vlues to nswer the questions elow y ompleting the sentene sin os tn undefined 8) s the ute ngle gets lrger, the sine of the ngle gets nd the osine of the ngle gets 9) The vlue of the sine nd osine of n ngle lwys lies etween the numers nd 0) The smllest vlue of the tngent of n ngle is nd the iggest vlue is [HINT: In the lst question, use your lultor to look t the vlues of tn 89, tn 89.9, tn 89.99, tn , tn , et] hpter : Right Tringle Trigonometry Pge

12 8..: USING TRIGONOMETRI RTIOS TO FIND N NGLE Most lultors work in one of two wys when omputing the ngle given the vlue of the trigonometri rtio. You need to e fmilir with yours. gin, first hek tht the MODE of the lultor is degrees [DEG] nd not rdins [RD] NOTTION: If If If sin x os x tn x then then then sin x os x tn x LULTOR TYPE LULTOR TYPE (older style). Press INV or ND utton. Enter the vlue of the rtio. Press SIN, OS or TN. Press INV or ND utton. Enter the vlue of the rtio. Press SIN, OS or TN. Press equl/enter to get the vlue Note: the equls sign is not needed. Round the resulting ngle to the required numer of deiml ples. Exmple : Find the ngle, orret to the nerest tenth of degree, given the rtio. () sin 0. () os 0. () tn θ. () If sin 0., find the ngle orret to the nerest tenth of degree, given the rtio. Enter: lultor Type lultor Type ND Use the INV or utton Disply: INV sin INV sin the numer of deiml ples given in the lultor s nswer will depend on your lultor: 7.99 Rounding: we wnt the nerest tenth of degree we pproximte orret to one deiml ple: nd sine the digit in the seond deiml ple is, whih is less thn, we leve the numer in the first deiml ple s. nswer: If sin 0. then 7. hek: Using the lultor, sin hpter : Right Tringle Trigonometry Pge

13 () If os 0., find the ngle orret to the nerest tenth of degree, given the rtio. Enter: lultor Type lultor Type ND Use the INV or utton Disply: INV os INV os the numer of deiml ples given in the lultor s nswer will depend on your lultor: Rounding: we wnt the nerest tenth of degree we pproximte orret to one deiml ple: Sine the digit in the seond deiml ple is 9, whih is more thn, we round the 9 in the first deiml ple to 0, whih will round 8.9 to 8.0. nswer: If os 0. then 8.0 NOTE: To orretly nswer this question, we should write 8.0 not 8 in order to indite tht this ngle is orret to the nerest tenth of degree. hek: Using the lultor, os () If tn θ., find the ngle θ orret to the nerest tenth of degree, given the rtio. NOTE: θ is the greek letter thet, ommonly used in trigonometry to represent n ngle. Enter: lultor Type lultor Type ND Use the INV or utton Disply: INV tn.. INV tn Rounding: Round the 8 in the first deiml ple to 9 nswer: If tn θ. then θ 7.9 hek: Using the lultor, tn This is not. s we would like. However, onsidering the lterntives, tn , tn nd tn , so we hve the losest pproximtion to one tenth of degree. hpter : Right Tringle Trigonometry Pge

14 Exerise 8.. Questions 8: Use lultor to find eh of the following. pproximte ll nswers s instruted. ), to the nerest degree, if sin ) θ, to the nerest tenth of degree if, tn θ. ) P, to the nerest degree if os P 0. 0 ) if os ) θ, to the nerest degree, if os θ ), if sin 7) θ, to the nerest tenth of degree, if tn θ ) if tn hpter : Right Tringle Trigonometry Pge

15 8..: SOLVING RIGHT TRINGLES The three sides nd three ngles of ny tringle re relted through the trigonometri rtios, nd we re le to use trigonometry to determine the mesure of the remining omponents if we re given ny three fts out the tringle. In this setion we re solving right tringles. Tht is, we know tht the tringle hs one ngle tht of 90. Given this nd: o the length of two sides, or o the size of one ngle nd the length of one side, we n find the remining sides nd ngles of the tringle. In hpter we will extend our theory to non-right tringles. Solve the Right Tringle Given One ngle nd One Side Exmple : Find the remining sides nd ngles of the tringle, given tht is right ngle, 0 nd 0 m. The first step is lwys to drw digrm of the tringle. Lel the digrm with the informtion given nd the unknowns to e found. In this exmple we know, nd. We need to find,, nd 0 0 To find we n use the ft tht the two ute ngles of the tringle re omplementry, To find nd we need to identify their reltive positions in trigonometri rtio of whih we know two out of the three omponents. To find, we n use either os (sine is the side to the given ngle, nd we 0 know the ) or sin (sine is the side to the lulted ngle, nd we know the ). 0 Using os : os Sustituting the known vlues for nd, os 0 0 Multiplying oth sides of the eqution y 0: 0os 0 lultor: 0 x os 0 Using lultor 7.0 Sine the lest numer of signifint figures in the dt is three, nswers in this prolem should e rounded orret to three signifint figures: 7. 7 hpter : Right Tringle Trigonometry Pge

16 LTERNTE: Find using sin : sin Sustituting the known vlues of nd : sin0 0 Multiplying oth sides y 0 : 0sin0 Using lultor : 7. 7 lultor: 0 x sin 0 To find, we n use either sin (sine is the side to the given ngle, nd we know the ) or os (sine is the side to the given ngle, nd we know the ), or we ould use Pythgors Theorem. Using sin : sin Sustituting the known vlues of nd : sin 0 0 Multiplying oth sides y 0 : 0sin 0 Using lultor :. LTERNTE: Find using os : os hptotenuse Sustituting the known vlues of nd : os0 0 Multiplying oth sides y 0 : 0os0 Using lultor :. lultor: 0 x sin 0 lultor: 0 x os 0 LTERNTE: We ould lso find using Pythgors Theorem. This is the lest ppeling method euse it uses n pproximted lultion rther thn n ext mesure. NOTE: While heking our lultions, notie tht the smllest side,. m, is the smllest ngle, 0 ; the middle side, 7.7 m, is the middle ngle, 0, nd the, 0 m, is the longest side of the tringle. hpter : Right Tringle Trigonometry Pge

17 Exmple : Solve the tringle, given is right ngle, 7. nd. units. To find : To find tn is hosen insted of tn only euse it results in n slightly esier mnipultion: tn To find : tn... tn. sin. sin. sin NOTE: Sine the lest numer of signifint figures in the dt is three, nswers in this prolem should e rounded orret to three signifint figures. lultor:. sin 7. Exmple : Find the ext vlues of the remining sides nd ngles of, given is right ngle, nd units. Sine one ngle of is we know tht the tringle is n isoseles tringle with two ngles of ; the sides these ngles re equl, nd oth units Therefore nd. In ft we lso know tht the sides re in the fixed rtio of :: nd do not need to use trigonometri rtios to find in this se. However to demonstrte the use of trigonometry, we will use the sine rtio to find. To find : sin sin sin We know sin, hene hpter : Right Tringle Trigonometry Pge 7

18 Solve the Right Tringle Given the Length of Two Sides Exmple : Solve the tringle, given is right ngle, m nd m. Drw digrm nd lel ll known nd unknown vlues. To find sin sin sin 7 ( ) The esiest method to find is to use the ft tht it is the omplement of lterntely, we n use the osine rtio: os lultor: INV sin ( ) os ( ) os lultor: INV os ( ) The esiest method to find is to use Pythgors Theorem or etter yet, to reognize the missing element of the {,, } Pythgoren Trid to find tht the ext mesure of is units. lterntely, we n use the sine, osine or tngent rtios to find : tn tn7 Rerrnging : tn7 This is the lest ppeling method euse we re using n pproximted lultion when the more urte given dt ould e used. hpter : Right Tringle Trigonometry Pge 8

19 Exmple : Solve the tringle, given is right ngle,. m nd. m. Sine we re given the nd sides for, we n use the tn rtio to find the ngle. tn. tn.. tn.. 70 Sine nd re omplementry: Using Pythgors Theorem, we n find We n use the tngent rtio to find : tn. tn.. tn Solve the Right Tringle Given Trigonometri Rtio Exmple : Find the sides nd ngles of tringle PQR, where P is right ngle nd sin R. 9 0 Sine sin R nd R 9 0 sin then we n use sin R q Tht is, sine we know tht the rtio of the sides is 9 to 0, we n set the side R s 9 units nd the of the tringle s 0 units. Drw digrm nd lel verties nd sides. Q y Pythgors Theorem, 9 + q 0 Sine sinr q 9 0 then, R sin 0 R ( 9 ) Q nd R re omplementry so: Q 90 Q 77 9 P q 0 R hpter : Right Tringle Trigonometry Pge 9

20 Exmple 7: In right tringle, where is right ngle, find os given tht tn. Sine tn nd tn then for in t this point, digrm is very useful! We need to find os. os Using Pythgors Theorem to find : + + Therefore os hpter : Right Tringle Trigonometry Pge 0

21 pplitions Exmple 9: Find the ltitude of n isoseles tringle whih hs se of length 8 m nd se ngles mesuring. We need to find h. Sine h is the side the ngle nd we hve the mesure of the side to we will use the tngent rtio. h h tn h tn Sine we need to pproximte orret to signifint figures, the height of the tringle is lulted s 0 m. Exmple 0: ft ldder lens ginst wll nd rehes to height of ft. Wht ngle does the ldder form with the wll? Drw digrm. Let θ e the ngle osθ θ os θ ft θ ft Therefore the ldder mkes ngle with the wll. Exmple : stellite is oriting 7 miles ove the erth s surfe. (See digrm) When it is diretly ove the point T, the ngle S is found to e 7.. Find the rdius of the erth. Let the rdius of the erth e r miles. Then: ( r + 7) r sin 7. r + 7 sin 7. r r sin sin7. r 7 sin7. r r sin 7. r ( sin 7. ) 7 sin7. r sin Rdius, r S 7 mi T r (r + ) mi r 00.8 Hene the rdius of the erth is pproximtely 000 miles (orret to three signifint figures.) hpter : Right Tringle Trigonometry Pge

22 ngles of Elevtion nd Depression ngles of Elevtion nd Depression re mesured reltive to the oserver. n imginry line drwn from the eye of the oserver to the ojet eing oserved is lled the line of sight. The horizontl is the line of sight to n ojet diretly in front of, neither ove nor elow, the eye of the oserver. If the ojet eing oserved is ove the horizontl, then the ngle etween the line of sight nd the horizontl is lled the ngle of elevtion. (nother wy to think of this is the ngle the oserver would need to look up to the ojet.) If the ojet is elow the horizontl, then the ngle etween the line of sight nd the horizontl is lled the ngle of depression. (nother wy to think of this is the ngle the oserver would need to look down to the ojet.) HORIZONTL ngle of Elevtion ngle of Depression Note tht the ngle of elevtion or depression is lwys mesured from the oserver to the ojet. HORIZONTL Exmple : To determine the height of tree, student oserves tht it sts ft shdow when the ngle of elevtion of the sun (from the top of the shdow) is. Find the height of the tree. We need to find h, the height of the tree. First drw digrm. h h The top of the tree, se of the tree/foot of the shdow, nd tip of the shdow form the three verties of right tringle. We know one ngle nd two sides of the tringle: the ngle of elevtion, the length of the shdow whih is n side of the ngle; nd wnt to find the height of the tree, whih is the side of the tringle to the ngle. Thus we use the tngent rtio. h tn h tn h 78 The tree is pproximtely 78 ft in height. hpter : Right Tringle Trigonometry Pge

23 Exmple : n irplne is flying t n ltitude of ft, diretly ove stright streth of highwy long whih r nd us re trveling towrds eh other. The vehiles re on sides of the irplne, the r t n ngle of depression of.7 nd the us t n ngle of depression of. from the plne. How fr prt re the vehiles to the nerest tenth of mile? Let the r e x ft nd the us e y ft from the point whih the irplne is flying diretly ove. Then the r nd the us re (x + y) ft prt. Sine we hve the mesure of the ngle nd its side, nd we wish to find the side, we use the tngent rtio. x tn. 7 x tn. 7 x.7. ft y y tn. y tn. x + y tn. 7 + tn. 0 ft ( tn. 7 + tn. ) 0 80 miles. 8 miles Hene the vehiles re pproximtely. miles prt. Exmple : From point.0 meters from the se of uilding, the ngle of elevtion to the top of the uilding is.. The ngle of elevtion from the sme point to the tip of flgpole on top of this uilding is 8.. Wht is the height of the flgpole? Let the uilding e h meters nd the flgpole e x meters in height. Then: h tn. h tn. nd x + h tn7. x + h tn8. x tn8. h Sustituting for h: x tn8. h tn8. tn ( tn8. tn. ) Hene the pole is pproximtely 9. m tll. 7. m hpter : Right Tringle Trigonometry Pge x h

24 Exerise 8... Solve the tringle, given tht is right ngle, nd: () 0 nd () 8 nd (). nd. (d) nd 7 (e) 8. nd 8.8 (f) nd. Find the vlue of the side leled x orret to the nerest tenth. () () () 7 x x.. x (d) (e) x 7 (f) 8 x 7. x. Find the ngle θ orret to the nerest degree. () () 8 () 8. θ θ θ. (d) (e) (f) θ 7. θ. θ. Solve the right tringle Q () () () 0 R Y X Z P. Solve for the ext vlues of the right tringle P () () R () 0 8 M 0 N S T hpter : Right Tringle Trigonometry Pge

25 . ldder 0 ft in length rehes 9 ft up wll ginst whih it lens. Find the ngle, to the nerest degree, tht the ldder mkes with the wll. 7. rod up hill mkes n ngle of. with the horizontl. If this rod is. miles long, how high is the hill, to the nerest hundred feet? 8. When the ngle of elevtion of the sun is 7 uilding sts shdow of ft. How tll is the uilding the nerest foot? 9. ft mn sts shdow tht is ft long. Wht is the ngle of elevtion of the sun, to the nerest degree? 0. The irle shown hs rdius of r, nd enter t. If the distne DE m, find the rdius of the irle to the nerest entimeter. r r r D E E F. The ue shown hs side length of 0 m. Find the ngle formed y the digonls G nd DG (orret to the nerest tenth of degree.) 0 H 0 G D 0. kite string is extended ft in length when the kite mkes n ngle of elevtion of. with the ground. Find the ltitude of the kite to the nerest foot. h ft.. To mesure the height of tower ross freewy, student tkes two mesurements. She stnds diretly ross from the point t the foot of the tower, nd finds tht the ngle of elevtion to the top of the tower is.. She then wlks 0 ft prllel to the freewy (t right ngle to the point t whih she took the mesure) nd then finds tht the ngle from her new lotion to the se of the tower is 87.. Using this informtion, find the height of the tower orret to the nerest foot. h ft hpter : Right Tringle Trigonometry Pge

26 . hot ir lloon is floting ove stright streth of highwy. To estimte how high ove the ground the lloon is floting, the pssengers of the lloon tke mesurements of r elow them. They ssume tht the r is trveling t 0 miles per hour. One minute fter the r psses diretly elow the lloon they tke ering on the r nd find tht the ngle of depression to the r is. Find the ltitude of the lloon to the nerest 00 ft. h. mn is stnding 0 ft from pinting. He noties tht the ngle of elevtion from his eyes to the top of the pinting is 8 nd the ngle of depression to the ottom of the pinting is. Find the height of the pinting to the nerest tenth of foot. 8 0 ft x. person stnding on hill sees tower tht she knows to e 0 ft high. She oserves tht the ngle of elevtion to the top of the tower is, while the ngle of depression to the foot of the tower is. How fr is she from the tower, orret to the nerest foot? x ft 0 ft 7. To estimte the height of prtiulr mountin, the ngle of elevtion to the top of the mountin is mesured to e ft loser to the mountin the ngle of elevtion is found to e. Wht is the height of the mountin to the nerest hundred feet? h 8 00 ft 8. Find the dimensions of the sheet of pper needed to drw n otgon of side m, to the nerest entimeter. hpter : Right Tringle Trigonometry Pge