Trigonometry. Trigonometry. Solutions. Curriculum Ready ACMMG: 223, 224, 245.
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1 Trgonometry Trgonometry Solutons Currulum Redy CMMG:, 4, 4
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3 Trgonometry Solutons Bss Pge questons. Identfy f the followng trngles re rght ngled or not. Trngles,, d, e re rght ngled ndted y the smll squre n the orner of eh trngle. d e. Wht s the sum of the nteror ngles of trngle? The sum of the nteror ngles of trngle s 80.. Identfy f the followng trngles re rght ngled. 0 Sum of the nteror ngles s 80. The lst ngle must e 90 for the trngle to e rght ngled. 40 Therefore, Therefore, the trngle s rght ngled sne the lst ngle s Therefore, Sum of the nteror ngles s 80. The lst ngle must e 90 for the trngle to e rght ngled Therefore, the trngle s not rght ngled sne the lst ngle s 00. Sum of the nteror ngles s 80. The lst ngle must e 90 for the trngle to e rght ngled Therefore, Therefore, the trngle s rght ngled sne the lst ngle s 90 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC
4 Trgonometry Solutons Bss Pge 4 questons 4. Lel the opposte, djent nd hypotenuse n eh of the followng trngles. pposte ypotenuse djent djent ypotenuse pposte djent pposte ypotenuse d djent ypotenuse pposte e pposte ypotenuse djent f pposte ypotenuse djent. Use the followng trngles to fll n the orret sdes n the tle elow: Trngle pposte to djent to pposte to djent to ypotenuse BC C BC BC C B DEF EF DE DE EF DF LMN MN LM ML MN LN PQR PR QR QR PR PQ WXY WX WY WY WX XY 00% Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
5 Trgonometry Solutons Knowng More Pge 9 questons. Use the trngles to omplete the tle elow: Trngle pposte to djent to ypotenuse sn os tn % Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC
6 Trgonometry Solutons Knowng More Pge 9 questons. Complete the followng for eh trngle: sn B 4 C 4 os + B tn + B D 0 F E os + E tn + D 0 sn + D L N 4 M tn + N sn + N 4 os + N 4 Pge 0 questons. Fnd the mssng sde n eh rght ngled trngle, nd then fnd the rtos tht follow: Fnd MN, use Pythgors MN MP + PN f MP, PN 8 M P 8 N MN MN MN MN! 00 MN! 0 MN 0 (snelengthspostve) sn + N 0 tn + M 8 4 os + M 0 tn + N % Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
7 Trgonometry Solutons Knowng More Pge 0 questons Fnd the mssng sde, use Pythgors + f, α !! sn tn sn os P Fnd PQ, use Pythgors QR PR + PQ f PR 8, QR 7 Q 7 8 R PQ PQ PQ 89-4 PQ PQ! PQ! PQ sn + Q 8 7 os + Q 7 os + R 8 7 tn + R 8 d Fnd ypoenuse, use Pythgors + f, 9 α ! 0 0 tn 9 os 0 sn 0 tn 9 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC
8 Trgonometry Solutons Knowng More Pge questons 4. Evlute the followng, to deml ples: sn ( d.p.) os ( d.p.) os 0 0. d tn ( d.p.) e tn 0.9 ( d.p.) f sn ( d.p.) g os 4. ( d.p.) h sn 4 tn 0 j 4sn 7 k os.8 ( d.p.) 0.40 ( d.p.) l tn ( d.p.) 4. Fnd the vlue of θ (to the nerest degree) f: os 0. sn 0. tn os - ( 0. ) sn - ( 0. ) tn - ( ) 0 4 (nerest degree) 0 d tn 4. e os 08. f tn 77 - ( 4. ) os - ( 08. ) (nerest degree) (nerest degree) sn sn - m 4 g os h tn m. os m - m tn (. ) (nerest degree) sn( ) - sn () % Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
9 Trgonometry Solutons Usng ur Knowledge Pge questons. Fnd n eh trngle to the nerest degree: 8 4. os os 8 os - 8 ` j 8 sn sn 4 sn - ` j 4 tn tn. tn - ` j. 4. Complete the tle elow f you re solvng sde leled : d Trngle Gven sde for ngle Mssng sde for ngle () Corret rto to use (sn, os, tn) djent ypotenuse os djent pposte tn ypotenuse pposte sn d djent ypotenuse os 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC 7
10 Trgonometry Solutons Usng ur Knowledge Pge questons. Fnd the vlue of n eh of the trngles from the prevous queston: 8 4 os tn 4 os 8 4 tn tn 4 os 8.4 ( d.p.) 7.8 ( d.p.) 4. 7 sn sn sn 7 d 4.7 os os os 4. ( d.p.).0 ( d.p.) 4. sker jumps 4m rmp. m fter the jump the sker's heght s m. Wht s the ngle of the rmp? Note: metres s dded to 4 sne the queston sttes " metres fter the rmp" therefore mkng the djent sde metres. 4 tn tn tn - ` j (nerest degree) 8 00% Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
11 Trgonometry Solutons Usng ur Knowledge Pge 7 questons. fshermn sts hs lne out nd keeps hs fshng rod pontng strght upwrds. If the lne touhes the wter 0 m from the shore t n ngle of 0, then how long s the fshng lne to the nerest metre? The length wnted s the hypotenuse. 0 0 os os os 0 4. m( d.p.) Therefore, the fshng lne s m.. If the fshng lne s 40 m long nd touhes the wter m from the shore, t wht ngle wll the lne touh the wter? 40 m os os 40 os - ` j 40 4 (nerest degree) 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC 9
12 Trgonometry Solutons Thnkng More Pge 0 questons. studo s 7 m to the left of shool. The ngle of elevton from the se of the studo to the roof of the shool s 44. The ngle of depresson from the roof of the studo to the roof of the shool s h 79 studo s shool 44 7 m Fnd the heght of the shool to deml ples. tn tn 44 s 7 s 7 # tn 44 s 70.49m( d.p.) ow muh hgher s the studo thn the shool to deml ples? tn tn 79 h 7 h 7 # tn 79 h 7.m( d.p.) Wht s the totl heght of the studo to deml ple? s+ h m( d.p.) 0 00% Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
13 Trgonometry Solutons Thnkng More Pge questons. skteorder reds sgn on top of 40 m uldng. Identfy the ngle of elevton nd the ngle of depresson n the followng dgrm: ngle of depresson 40 m 80 m ngle of elevton If he sees the sgn when he s 80 m wy from the uldng, wht s the ngle of elevton from the skter to the sgn? tn tn tn - 40 ` j 80. ( d.p.) If the skter ontnues sktng untl he s 0 m from the uldng, wll the ngle of elevton nrese or derese? By how muh? tn tn 40 0 tn - 40 ` j 0. ( d.p.) The ngle of elevton wll nrese. It wll nrese y.. 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC
14 Trgonometry Solutons Thnkng More Pge questons. den nswered the followng queston norretly. Cn you spot hs mstke? The ngle of depresson from helopter to ts lndng se s. If the horzontl dstne etween the helopter nd the lndng se s 0 m, then how hgh s the helopter ( deml ple) t ths pont? IDEN S SLUTIN h 8 tn % 0 h h 0 tn % h m se h m ( deml ple) Wht ws den's mstke? den norretly lelled the ngle of depresson. The ngle of depresson s formed etween the upper horzontl nd hypotenuse not h nd the hypotenuse. Fnd the orret heght of the helopter t ths pont. If the ngle of depresson s then the ngle etween the hypotenuse nd h s 90- whh s 8. tn tn 8 0 h h 0 tn 8 h 9.0m( d.p.) 00% Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
15 Trgonometry Solutons Thnkng More Pge questons 4. n eroplne tkes off t n ngle of 8 to the ground. It fles over house 900 m from the rport. ow hgh s the eroplne t tht pont, to deml ples? 8 h tn tn 8 h 900 h 900# tn m h m( d.p.) Wht s the ngle of depresson t ths pont? The ngle of depresson s 8. fter ontnung to fly t the sme heght, the plot notes tht s they re flyng over lke, the rport hs ngle of depresson. ow fr s the lke wy from the rport, to deml ples? tn tn tn 78.9m( d.p.) 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC
16 Trgonometry Solutons Thnkng More Pge 4 questons. stellte tower s on the rght of post offe nd they re seprted y dstne, d. The post offe hs heght of m. The ngle of depresson from the roof of the post offe to the se of the tower s. The ngle of elevton from the roof of the post offe to the roof of the tower s 8. Drw dgrm to represent ths stuton. m Post offe 8 Stellte tower d Fnd d, the dstne etween the uldngs to deml ple: Fnd, 90-7 tn tn 7 d d # tn 7 d 8. m( d.p.) Fnd the totl heght of the tower to deml ple: tn tn 8 8. h 8. # tn 8 h 4. m( d.p.) Therefore, the totl of heght s equl to 4. m+ m 7. m 4 00% Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
17 Trgonometry Solutons Thnkng Even More Pge 7 questons. You nd your frend stnd n uldng wth 0 floors; eh floor s m hgh. You re on the 4 th floor nd your frend s on the top floor. Fnd the dfferene etween your ngles of elevton 0 m wy. 0 top floor 4 th floor Fnd, To fnd the opposte: # 0 00 m sne eh floor s m nd here re 0 floors. tn tn 00 0 tn - 00 ` j 0 9 To fnd the opposte: # 4 8 m sne eh floor s m nd here re 4 floors. tn 8 0 tn tn The dfferene s: s tehnn you need to te rope long the dotted lne n ths retngle. ow mny rght ngled trngles re nvolved n ths prolem? 8 0 There re rght ngled trngles. owever, t s only neessry to use one rght ngled trngle wth the nformton gven, to fnd ll neessry lengths. 00% Trgonometry Solutons Mthlets 00% P Lernng SERIES TPIC
18 Trgonometry Solutons Thnkng Even More Pge 7 questons Fnd the totl length of rope needed f ll mesurements re n m (nerest m): sn sn 0 0 sn 7.4 m( d.p.) tn tn 0 0 tn 4. m( d.p.) Therefore the totl length of rope needed s 7.4 m+ 4. m.7 m whh rounds to m Pge 8 questons. In order to for ertn kte to fly t needs to look lke ths. Fnd the length of B nd ngle + BE eh to deml ple. D 0 E m sn sn 0 E E # sn 0 E 9.9 m( d.p.) C 4 m Usng Pythgors, B B E + EB B B ( d.p.) B 804. B.7 m( d.p.) tn + B BE tn ( 04. ) + BE. ( d.p.) 00% Trgonometry Solutons SERIES TPIC Mthlets 00% P Lernng
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20 Trgonometry
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