Trigonometry. Trigonometry. Curriculum Ready ACMMG: 223, 224, 245.

Transcription

1 Trgonometry Trgonometry Currulum Rey ACMMG: 223, 22, 2

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3 Trgonometry TRIGONOMETRY Bslly, mny stutons n the rel worl n e relte to rght ngle trngle. Trgonometry souns ffult, ut t s relly just methos to fn the se lengths n ngle szes n these trngles. Answer these questons, efore workng through the hpter. I use to thnk: Wht s the longest se of trngle lle? Wht re the other ses lle? Eh ngle hs 3 mn trgonometr rtos? Wht s trgonometr rto? Wht re the 3 mn rtos? When woul you use the nverse of rto? Answer these questons, fter workng through the hpter. But now I thnk: Wht s the longest se of trngle lle? Wht re the other ses lle? Eh ngle hs 3 mn trgonometr rtos? Wht s trgonometr rto? Wht re the 3 mn rtos? When woul you use the nverse of rto? Wht o I know now tht I n t know efore? 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 1

4 Trgonometry Bss Rght Angle Trngle A rght ngle trngle s trngle whh hs n ngle of Rght ngle NOT rght ngle Rght ngle 0 Do you rememer the sum of the nteror ngles of trngle? The longest se of rght ngle trngle s lle the hypotenuse. It s lwys the se opposte the rght ngle. Hypotenuse Opposte n Ajent ses to The Greek letters (thet) n α (lph) re use to lel ngles. The ses of trngle re lele s ether jent (next to) or opposte these ngles. Ajent to α Opposte to Hypotenuse Opposte to α Hypotenuse Ajent to Opposte to α Ajent to 2 100% Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

5 Trgonometry Questons Bss 1. Ientfy f the followng trngles re rght ngle or not? e f 2. Wht s the sum of the nteror ngles of trngle? 3. Ientfy f the followng trngles re rght ngle or not? % Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 3

6 Trgonometry Questons Bss. Lel the opposte, jent n hypotenuse n eh of the followng trngles. e f. Use the followng trngles to fll n the orret ses n the tle elow: A α D E α N L C B α M P α Q F W α X R Y Trngle Opposte to Ajent to Opposte to Ajent to Hypotenuse ABC AC BC AB DEF DE EF LMN LM LN PQR QR WXY 100% Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

7 Trgonometry Knowng More Sn ABC s rwn nse ADE. The two trngles re smlr. Ths mens they hve equl ngles (n you show ths?) E Rememer C Hypotenuse Ajent Opposte 7.2 A 3 B 2. D We're gong to fn the vlue of the rto In ABC (smll trngle): opposte seto+ A BC hypotenuse AC 0.8 opposte seto+ A for oth the smll n g trngles. hypotenuse In ADE (g trngle): opposte seto+ A DE hypotenuse AE Ths mens the rto oes not epen on the sze of the trngle sne the rto hs the sme vlue for fferent trngles. The rto only epens on the ngle. Ths rto s lle sn (pronoune: sne thet). The formul for sn s: sn opposteseto hypotenuse Look t the followng exmples: Fn sn n sn 13 α opposte sn hypotenuse 13 opposte sn hypotenuse % Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC

8 Trgonometry Knowng More Sn, Cos n Tn (Trgonometr Rtos) Hypotenuse Opposte to Ajent to We lrey know tht for ny rght ngle trngle, the sne rto of n ute ngle s: sn opposte hypotenuse Hypotenuse Opposte to Ajent to sn wll e the sme for ny sze rght ngle trngle sne the sne rto only epens on the ngle. A seon rto s lle the osne rto. We wrte ths s os n t s gven y: os jent hypotenuse Hypotenuse Opposte to Ajent to os wll e the sme for ny sze rght ngle trngle sne the osne rto only epens on the ngle. The thr rto s lle the tngent rto. We wrte ths s tn n t s gven y: tn opposte jent Hypotenuse Ajent to Opposte to tn wll e the sme for ny sze rght ngle trngle sne the tngent rto only epens on the ngle % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

9 Trgonometry Knowng More Look t these exmples: Fn sn, os n tn n the followng trngle: Hypotenuse 17 8 Opposte 1 Ajent opposte sn hypotenuse os jent hypotenuse tn opposte jent Fn sn, os n tn n the followng trngle: Ajent to 12 Hypotenuse 1 9 Opposte sn opposte hypotenuse os jent hypotenuse tn opposte jent % Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 7

10 Trgonometry Knowng More Clultor Trks Your lultor hs uttons lle sn, os n tn. Rememer, ll of these only epen on the ngle. Mke sure your lultor s set to egree moe. To fn the vlue of ny rto just press the uttons: Rto Angle Fn the followng rtos: sn 7 os67 sn 7 os tn18 2 os 67 tn 18 2 x os Aove s the metho to fn the vlue of the trgonometr rto from the ngle. A lultor s lso use to fn the ngle from the trgonometr rto. To fn the vlue of the ngle, just press the shft key. Here re some exmples: Fn f: os 1 tn 1 2 shft os 0. shft tn sn 07. shft sn 0.7 shft os 0.8 os (nerest egree) 37 (nerest egree) 8 100% Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

11 Trgonometry Questons Knowng More 1. Use the trngles to omplete the tle elow: Trngle Opposte to Ajent to Hypotenuse sn os tn Complete the followng for eh trngle: A D 11 6 C 2 B 10 F N sn + A os tn + N os tn + D sn tn + B sn 11 os 221 E L 3 M % Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 9

12 Trgonometry Questons Knowng More 3. Fn the mssng se n eh rght ngle trngle, n then fn the rtos tht follow: M 6 12 α 13 P 8 N sn + N sn tn + M tn os + M sn tn + N os P 8 α Q 17 R 9 sn + Q tn os + Q os os + R sn tn + R tn % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

13 Trgonometry Questons Knowng More. Evlute the followng, to 3 eml ples: sn0 os30 os60 tn20 e tn0 f sn8 g 3os h 2 sn 3 tn30 j sn73 k os 23 % 2 l 3tn 80 %. Fn the vlue of (to the nerest egree) f: os 0. sn 0.2 tn 3 tn. e os 0.81 f sn 2 2 g os 3 2 h tn 2 m 3.1 sn(2) 1 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 11

14 Trgonometry Usng Our Knowlege Fnng Angles n Rght Trngles We use the shft utton to fn ngles n trngles. To wrte the nswers esly, use the formuls: sn os tn shft shft shft sn os tn Fn n the followng trngle: 10 Sne the opposte se n the hypotenuse re gven, t mkes sense to use sn. opposte sn hypotenuse 10 sn As efore: shft sn 0. sn The frst step s lwys to fgure out whh trgonometr rto mkes the most sense for the ngle, se on wht s gven. Fn n n the followng grm to the nerest egree: jent os hypotenuse 7 opposte tn jent α As efore: 1 os - ` 7 j. 1 As efore: 1 tn - 7 ` 3 j (nerest egree). 67 (nerest egree) For, n the ove exmple, t me sense to use os sne the jent n hypotenuse were gven for. It me sense to use tn sne the opposte n jent were gven for % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

15 Trgonometry Usng Our Knowlege Fnng Ses n Rght Trngles If we re gven se n n ute ngle of rght trngle, then we n fn mssng ses usng trgonometry rtos. In eh trngle we wll e gven n ngle n e gven ether the hypotenuse, opposte se or jent se. Fn the lengths of x n y n the followng trngle (to 2 eml ples) y x 8 1 Use sn1 to fn y sne t s the opposte se n we know the hypotenuse. y sn 1 8 y 8# sn 1 8# ^ h ( 2 eml ples) Use os1 to fn x sne t s the jent se n we know the hypotenuse. x os 1 8 x 8# os 1 8# ^ h ( 2 emlples) In these type of questons, hoose the rto whh nvolves the gven se n the mssng se. Aove sn ws hosen to solve for y euse t nvolves the opposte se (neee) n the hypotenuse (gven). Fn the length of the hypotenuse elow (to 2 eml ples) 12 h Choose sn sne t nvolves the opposte se (gven) n the hypotenuse (neee). sn 12 h h 12 sn ( 2 eml ples) 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 13

16 Trgonometry Usng Our Knowlege In orer for kte to fly, t nees to hve the ngles n the grm. DE s 32 m. Fn the lengths of AD & EC: For AD, use ADE. Sne we re mssng the hypotenuse AD n the jent se to +ADE hs een gven s 32m, we use os. A os jent hypotenuse DE AD os + ADE D 68 32m E B 32 os 68 AD AD 32 os 68 Mke AD the sujet C m ( 2 eml ples) For EC, use 3 CDE. Sne we re mssng EC the jent to +ECB n the opposte se to +ECB hs een gven s 32 m, we use tn. opposte tn jent DE EC tn + ECD 32 EC tn 2 % EC 32 tn 2 % Mke EC the sujet m ( 2 eml ples) 1 100% Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

17 Trgonometry Questons Usng Our Knowlege 1. Fn n eh trngle to the nerest egree: Complete the tle elow f you re solvng se lele x: x x Hypotenuse Ajent Opposte 1 x.2 17 x Trngle Gven se for ngle Mssng se for ngle (x) Corret rto to use (sn, os, tn) jent os opposte hypotenuse 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 1

18 Trgonometry Questons Usng Our Knowlege 3. Fn the vlue of x n eh of the trngles from the prevous queston:. A sker jumps m rmp. 2m fter the jump the sker's heght s 12m. Wht s the ngle of the rmp? % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

19 Trgonometry Questons Usng Our Knowlege. A fshermn sts hs lne out n keeps hs fshng ro pontng strght upwrs. If the lne touhes the wter 30 m from the shore t n ngle of 30, then how long s the fshng lne to the nerest metre? If the fshng lne s 0 m long n touhes the wter 33 m from the shore, t wht ngle wll the lne touh the wter? 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 17

20 Trgonometry Thnkng More Angles of Depresson n Elevton These ngles re lwys me wth horzontl lne. Angle of Elevton s the ngle lookng up Angle of Depresson s the ngle lookng own Horzontl Angle of epresson Angle of elevton Horzontl These ngles wll lwys hve the sme vlue, even though they re n fferent ples. (Do you know why?) A Ferrs wheel hs mxmum heght of 60 m n sts show 100 m long Wht s the ngle of elevton from the tp of the show to the top of the ferrs wheel? (to the nerest egree) tn tn tn -1-1 ` j 60 m α - tn 1 shft tn 100 m % %. 31 (nerest egree) Wht s the ngle of epresson,, from the top of the ferrs wheel to the tp of the show? The ngle of epresson hs the sme vlue s the ngle of elevton % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

21 Trgonometry Thnkng More When workng wth these questons, the key s to etermne the poston of the rght ngle trngle n the grm. The ngle of epresson from the top of lghthouse to shp s 60. The lghthouse s 1 m wy from the shp. Drw grm to represent ths stuton: 60 1 m If the grm s rwn orretly, then mssng ses n ngles n e use n the ext sme esy wy s n the prevous seton. A photogrpher stns on the ege of 92 m lff n tkes photo of flower. If the ngle of epresson of the mer s 68, then wht s the stne,, etween the lff n the flower? (nerest metre) Metho 1 Metho m 92 m 68 The ngle wth the vertl s Angle of elevton Angle of epresson 68. So, tn # tn # So, tn tn m (nerest egree). 37 m (nerest egree) 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 19

22 Trgonometry Questons Thnkng More 1. A stuo s 73 m to the left of shool. The ngle of elevton from the se of the stuo to the roof of the shool s. The ngle of epresson from the roof of the stuo to the roof of the shool s h 79 stuo s shool 73 m Fn the heght of the shool to 3 eml ples: How muh hgher s the stuo thn the shool to 3 eml ples? Wht s the totl heght of the stuo to 1 eml ple? % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

23 Trgonometry Questons Thnkng More 2. A skteorer res sgn on top of 0 m ulng. Ientfy the ngle of elevton n the ngle of epresson n the followng grm: 0 m 80 m If he sees the sgn when he s 80 m wy from the ulng, wht s the ngle of elevton from the skter to the sgn? If the skter ontnues sktng untl he s 30 m from the ulng, wll the ngle of elevton nrese or erese? By how muh? 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 21

24 Trgonometry Questons Thnkng More 3. Aen nswere the followng queston norretly. Cn you spot hs mstke? The ngle of epresson from helopter to ts lnng se s 2. If the horzontl stne etween the helopter n the lnng se s 10 m, then how hgh s the helopter (1 eml ple) t ths pont? AIDEN S SOLUTION 2 h tn 2 % 10 h h 10 tn 2 % h m se h m ( 1 eml ple) Wht ws Aen s mstke? Fn the orret heght of the helopter t ths pont % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

25 Trgonometry Questons Thnkng More. An eroplne tkes off t n ngle of 28 to the groun. It fles over house 900 m from the rport. How hgh s the eroplne t tht pont, to 3 eml ples? Wht s the ngle of epresson t ths pont? After ontnung to fly t the sme heght, the plot notes tht s they re flyng over lke, the rport hs 1 ngle of epresson. How fr s the lke wy from the rport, to 2 eml ples? 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 23

26 Trgonometry Questons Thnkng More. A stellte tower s on the rght of post offe n they re seprte y stne. The post offe hs heght of 12 m. The ngle of epresson from the roof of the post offe to the se of the tower s 23. The ngle of elevton from the roof of the post offe to the roof of the tower s 8. Drw grm to represent ths stuton: Fn, the stne etween the ulngs to 1 eml ple: Fn the totl heght of the tower to 1 eml ple: 2 100% Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

27 Trgonometry Thnkng Even More Prolems Involvng More Thn 1 Trngle As ll mthemtns know, most prolems n rel lfe re more omplte. They oul nvolve more thn 1 trngle nternlly or externlly. Let s sy you hve to e t ulng (whh s 83 m tll) n 1 hour. The ngle of elevton from you to the top of the ulng s 11. After 0 mnutes of wlkng loser to the ulng, the ngle of elevton hs nrese to 38. NOT TO SCALE A 83 D 11 C 38 B (Before wlkng) (After 0 mnutes) Whh re the two rght ngle trngles nvolve n ths prolem? ABD n ABC (Hghlght these) How fr were you from the ulng efore you strte wlkng (nerest metre)? tn 11 % AB 83 BD BD BD 83 tn11 % m (nerest metre) If you keep wlkng t the sme pe, re you gong to mke t to the ulng n tme? Frst we fn BC: tn 38 % AB BC BC AB m tn 38 % (nerest metre) Ths mens tht the stne wlke n 0 mnutes DC 27 m m 321 m. Ths s 8.02 m per mnute. Thus n 20 mnutes you woul wlk 20 # 8.02 m 160. m. Sne you only hve to wlk 106 m n 20 mnutes, you woul mke t on tme. 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 2

28 Trgonometry Thnkng Even More Most prolems nvolvng more thn 1 trngle wll sk you to fn ommon se (or ngle) of two trngles n then use tht ommon se (or ngle) to fn somethng (se or ngle) n seon trngle. A gnt gte nees to e ult n the shpe elow (ll mesurements n m) A B C E D Fn the heght of the gte: To fn the heght of the gte we nee AE or BD. BD AD os 60 BD AD # os # 0. m Fn the vlue of to 1 eml ple: sn BD 062. CD 8 1 sn - ^062. h ( 1 eml ple) In the exmple ove, we h to fn the ommon se BD usng wht we knew out 3 ABD to fn the ngle n 3 BCD % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

29 Trgonometry Questons Thnkng Even More 1. You n fren stn n ulng wth 0 floors, eh floor s 2 m hgh. You re on the 3 th floor n your fren s on the top floor. Fn the fferene n your ngles of elevton from 60 m wy. top floor 3 th floor As tehnn you nee to te rope long the otte lnes n ths retngle: How mny rght trngles re nvolve n ths prolem? Fn the totl length of rope neee f ll mesurements re n m (nerest m): 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 27

30 Trgonometry Questons Thnkng Even More 3. In orer for ertn kte to fly t nees to look lke ths. Fn the length of AB n ngle + ABE eh to 1 eml ple. D 0 12 m A E C 1 m B % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

31 Trgonometry Answers Bss: Knowng More: Trngles,,, e re rght ngle nte y the smll squre n the orner of eh trngle. The sum of the nteror ngles of trngle s s rght ngle s not rght ngle s rght ngle 1. Hypotenuse sn os tn Ajent to Trngle Opposte to Opposte Hypotenuse Ajent Ajent Hypotenuse Opposte 2. A 18 C 2 30 B sn + A os + B tn + B 3 Ajent Opposte Hypotenuse Ajent Hypotenuse Opposte D F 11 E os + E tn + D sn + D e Trngle Opposte Hypotenuse Opposte to Ajent Ajent to f Opposte to Opposte Hypotenuse Ajent to Ajent Hypotenuse L 6 N 3 M tn + N 1 2 sn + N 3 os + N 6 ABC AC BC BC AC AB DEF EF DE DE EF DF LMN MN LM ML MN LN PQR PR QR QR PR PQ WXY WX WY WY WX XY 100% Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 29

32 Trgonometry Answers Knowng More: Usng Our Knowlege: 3. MN sn + N 3 os M 3 + tn M 3 + tn + N 3 2. Trngle Gven se for ngle Mssng se for ngle (x) Corret rto to use (sn, os, tn) Ajent Hypotenuse os sn sn 13 tn 12 os 13 Ajent Opposte tn Hypotenuse Opposte sn Ajent Hypotenuse os PQ 1 sn + Q 8 os + Q os + R 8 tn + R ( 1.p.) 7.8 ( 1.p.) 1.2 ( 1.p.) 63 (nerest egree) 1.0 ( 1.p.) m (nerest egree) tn sn os tn Angle 3 (nerestegree) Thnkng More:. sn os s 70. 9m( 3.p.) os tn h 37.2m( 3.p.) e tn f sn m( 1.p.) g k 3os h 2 sn 1 3 tn 30 1 j sn os l 3tn m Angle of epresson 80 m Angle of elevton 77 e 36 f 26.6 ( 1.p.) g 30 h 1 The ngle of elevton wll nrese. It wll nrese y 26.. (1.p.) % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

33 Trgonometry Answers Thnkng More: 3. Aen norretly lelle the ngle of epresson. The ngle of epresson s forme etween the upper horzontl n hypotenuse not h n the hypotenuse. h 192.0m( 1.p.) m( 3.p.) Angle of epresson m( 2.p.). 12 m Post offe 8 23 Stellte tower 28.3 m( 1.p.) Totl heght 7.3 m (1.p.) Thnkng Even More: 1. Dffereneofngles 10. ( 1.p.) 2. There re 3 rght ngle trngles. Totl length of rope 32 m 3. AB m + ABE % Trgonometry Mthlets 100% 3P Lernng SERIES TOPIC 31

34 Trgonometry Notes % Trgonometry SERIES TOPIC Mthlets 100% 3P Lernng

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36 Trgonometry