Appendi E: Mnipuling Fions Te ules fo mnipuling fions involve lgei epessions e el e sme s e ules fo mnipuling fions involve numes Te fundmenl ules fo omining nd mnipuling fions e lised elow Te uses of ese ules e illused moe ompleel le in is ppendi ) Adding fions d + + d d Suing fions d d d ) Mulipling fions d d d) Dividing fions d d d Inve nd mulipl Numeo nd denomino Fions epess io of wo quniies Fo emple, e fion epesses e io of quni o quni Te quni ppes on e op of e fion is lled e numeo In is se, e numeo is Te quni ppes on e oom of e fion is lled e denomino In is se, e denomino is Finding ommon denomino Adding nd suing fions usull equies ommon denomino, is, ll of e fions involved ve e sme denomino Emple E Find ommon denominos fo e following olleions of fions Epess e fions using is ommon denomino
),, ), + + + Soluion: ) Te ommon denomino is Te wo fions n e epessed mulipling o numeo nd denomino (op nd oom) weve fo is needed o onve e denomino o e ommon denomino A ommon denomino n lws e oined jus mulipling e of e denominos ogee A possile ommon denomino is ( ) Epessing e wo fions using is ommon denomino: Noe, oweve, ( ) Te signifine of is osevion is sine led s e denominos of o nd s fos So, is led ommon denomino fo o of ese fions nd ould e used insed of ) One w o oin ommon denomino is o jus mulipl e denominos of e wo fions, + + + ogee Howeve, + + ( + ) Sine + is fo of + +, + + n e used s ommon denomino Epessing e wo fions using e ommon denomino + + : + +
+ + + + + + + Te dvnge of using + + s ommon denomino (e n, s, e moe ovious ( + ) ( + + ) + + + ) is e fions ou oin using e ommon denomino e usull e simples possile Adding nd suing fions Fions n onl e dded nd sued wen e wo fions ve e sme denomined If e wo fions do no ve e sme denomino, en ommon denomino mus e found efoe e fions n e dded o sued Te neessi of finding ommon denomino is w e podu of e wo denominos (ie d) ppes in e ules epessed elow Epessed using lgei smols, e ules fo dding nd suing funions e: d d + + + d d d d d d + d d d d Emple E Evlue nd simplif e of e lgei epessions ) + + ) + + + d) Soluion: ) + + + + + 5 6 + 4+ + + + + + + + + ( ) + ve e sme denomino) (Te wo fions led
) + + + +( + ) + ( + ) + + + + + ( + ) ( + ) d) + + + + + + + + 5 6 4 + ( + ) + ( + ) ( + )( + ) ( + ) + ( + ) ( + ) ( + ) ( + ) Some fue simplifiions e possile in P (d) Fo emple, ou ould mulipl ou ll of e kes Te impon poin in P (d), oweve, is using e simples (o les ) ommon denomino, e lgei fom of e esul will e s simple s possile In e se of P (d), e simples ommon denomino is ( + )( + )( + ), e n e moe ovious ommon denomino ( + )( + )( + )( + ) Mulipling fions Mulipling wo fions is peps e mos sig-fowd of ll opeions You simpl mulipl e numeos nd mulipl e denominos Epessed using lgei smols, is ule is: d d Emple E Evlue nd simplif e following fions s mu s possile ) + + + + sin θ os θ sin θ os θ + θ θ Soluion: ) + + ( + ) + + / + + + + + sin θ os θ sin θ os θ sin θ sin θ os θ os θ + θ θ θ Dividing fions Mn people ememe e ule fo dividing one fion noe ememeing ou mus inve e denomino nd mulipl Epessed s lgei smols, e ule is:
d d d Emple E4 Evlue o simplif e of e fions given elow ) + 4 / ) + + Soluion: ) 4 + + + 4 / / ) + + + + + + Simplifing omplied fions Someimes, fions will ve oe fions in ei numeo o denomino (o o) To simplif nd evlue omplied fions, evlue e numeo nd denomino sepel, nd en divide e wo Emple E5 Simplif e of e omplied fions s mu s possile
) + / + + + + ) + + 9 + 9 Soluion: ) + + + ( + ) ( + ) ( + ) / + + + + ( + ) + + + + + + + + + + + + / ( + ) + + ) + + 9 + 9 + + + 9 + + 9 ( + + ) ( 9 ) ( + ) + 9 9 + + + + 4 Eeises fo Appendi E Fo Polems -0, evlue e quni wiou using lulo 7 + 9 + 4 4 + 8
5 7 6 7 9 7 8 7 9 7 9 z z 0 + + Fo Polems -0, pefom e opeion indied Simplif ou nswes s mu s possile L+ w+ L L+ w ( ) + 4 + 7 + 5 q q + q+ 6 + 7 4 7 + /
8 + + + + 9 + + 0 sin θ os θ sin θ os θ + + Fo Polems -5, deide wee e of e semens e ue o flse + + 4 + + + + 5 + + + + + Answes o Eeises fo Appendi E / /6 /4 4 (6 + )/(4) 5 (4 - )/(7) 6 ()/(6 7 7/( ) 8 / 9 (z)/ 0 /( - ) (L + wl + w + L)/(L + w)
( - /( - 4 ( + )/(7 + ) 5 (q - )/(q + ) 6 (9-8)/(7-8) 7 ( + / + -/ )/( - ) 8 / 9 / 0 sin(θ ) + os(θ) Flse Tue Flse 4 Flse 5 Flse