Chpte Seven Notes N P UC7 Nme Peiod Setion 7.: Angles nd Thei Mesue In fling, hitetue, nd multitude of othe fields, ngles e used. An ngle is two diffeent s tht hve the sme initil (o stting) point. The s e lled the sides of the ngle, nd the initil point is lled the vete of the ngle. In the ngle shown to the left, the vete of the ngle is S. the sides of the ngle e SR nd ST. the ngle n e lled RST, TSR, o S. One of n ngle is the initil side nd the othe is the teminl side. Two tpes of ngles n e fomed sed on the dietion nd mount of ottion fom the initil side to the teminl side. If the ottion is in the lokwise dietion, the ngle is negtive; if the ottion is ountelokwise, the ngle is positive. While we will e using positive ngles most of the time, we do use negtive ngles quite it s well. Howeve, one es w to get negtive ngle k to positive ngle is dding 0, o moving n entie ile (o moe thn one ile) ound. Emple : Wht would e the positive ngle ssoited with the following ngles? ) 7: 7 + 0 = 7 is the sme s the positive ngle. ) 777: 777 + 0 = 7 7 + 0 = 57 57 + 0 = 0 777 is the sme s the positive ngle 0. pge N P UC7
A new w of mesuing ngles involves dins. You lened k in geomet tht entl ngle is n ngle tht is positive nd tht hs its vete t the ente of the ile. The s of tht entl ngle inteset n on the ile, o sutends tht. If the sutended s is the sme s the dius, the ngle tht inteepts tht is one din, o = din. To find the length fo something othe thn one din, ou n use fomul o ou n use some ommon sense mth. To get the fomul, ou ould set up popotion whee din omped to its length of (sine the length would hve to e dius fo din) would oespond to nothe ngle with its length of s. Cossmultipling, ou get tht s =. s Fomul fo length: s = (s is the length, is the dius, nd is ngle in dins) Knowing this, ou ould elize tht the entie ile s length, onsideed s n s, is the iumfeene of the ile, o. If ou wnted to find the dins ssoited with n entie ile, knowing tht s =, ou ould sustitute this into the fomul fo length. s = = = Fomul fo length Sustitution Dividing on oth sides This mens tht ile, o one evolution, is mde up of dins. Going k to geomet (nd me even elie), ou ememe tht ile, o one evolution, is mde up of 0. Keeping tht in mind, dins = 0 dins = 80 Dividing on oth sides Using this eltionship, ou n onvet etween dins nd degees. Emple : Convet the following degees into dins. Give et, simplified nswes in tems of whee ppopite. pge N P UC7
) 5: 5 dins (using dins fo eve 80 to get degees to nel out) 80 5 dins dins 80 5 is the sme s dins. ) 58: 58 dins (using dins fo eve 80 to get degees to nel out) 80 58 dins 9 dins 80 90 58 is the sme s 9 dins. 90 Emple : Convet the following dins into degees. Round nswes to hundedths. ) dins: dins 80 (using dins fo eve 80 to get dins to nel out) dins 0 0 0. 00 dins is the sme s 0, o 0.00 (to the hundedths). ) dins: dins 80 (using dins fo eve 80 to get dins to nel out) dins 50 7. 89 dins is the sme s ppoimtel 7.89 (to the hundedths). Using these eltionships, ou n epnd to finding the length of n o even the e of ile oveed etin entl ngle. Emple : An of length 0 m is fomed on ile with 5 m dius. Wht is the ngle (in degees) whih it is sutended (inteepted)? Thee e two ws to do this. ) Fomul: The fomul to find length is s =. You know tht s is 0 m nd is 5 m, so ou n sustitute nd solve. s = 0 m = (5 m) = dins (sine HAS to e in dins to use this fomul) Sine the question sks fo the ngle to e in degees, ou must onvet dins into degees. pge N P UC7
dins 80 (using dins fo eve 80 to get dins to nel out) dins 0. 59 ) Geomet: Rell k to geomet tht the iumfeene of ile is C =. If ou wee to tke ftion of tht ile, nmel ftion of (o 0), ou ould find the ngle s well dietl in degees. In othe wods, if totl ile is C, o (o 0), ftion of the ile s would e, o. 0 0 m (5 m) (0 m is ftion of full ile with dius of 5 m) 0 0 m 0 m 0 (Multiplition) 0 (Dividing 0 on oth sides) 0 (Cossmultipling) 0. 59 (Dividing on oth sides) The eson I pesented the seond w to do this is tht ou n use the sme onept to find the e of pt of ile known s the seto of ile. Rell fom geomet tht A. If ou wnted onl ftion of tht full ile, o seto of tht ile, using entl ngle in dins, ou would stte tht the ftion of tht ile ou wnt would e. Appling this to the e fomul to find the e of the seto: A Fomul fo seto e: A (A is the seto e, is the dius, nd is ngle in dins) Agin, this ONLY woks in dins, ut, if ou just tke the ftion dins o 0 of the e if the ngle is in if the ngle is in degees, ou don t hve to memoize nothe fomul. pge N P UC7
Emple 5: A iul pond t pk hs designted fishing e mked with two opes tthed to uo t the ente of the pond with eh ope tking up 9 ds in length. Togethe, the two opes fom n ngle of 70. Wht is the e designted fo fishing? Thee e two ws to do this. You n eithe use fomul, elizing ou will hve to onvet degees to dins efoe using it, o ou n tke ftion of the totl e of the ile. ) Fomul: The fomul to find seto e is A. You know tht is 9 ds, ut ou will fist hve to onvet the 70 into dins. 70 dins (using dins fo eve 80 to get degees to nel out) 80 70 dins 7 dins 80 8 7 Knowing tht is now dins, ou n use the fomul. 8 A (seto fomul) 7 A 9 d 0.7 d 8 The designted fishing e is ppoimtel 0.7 sque ds. ) Geomet: The designted fishing e is 70 out of the totl 0 of the pond, so let s tke ftion of tht e. 70 A 0 (Tking the ftion of the totl e of the pond) 70 A 9 0.7 d 0 (Sustituting nd lulting) Agin, we still find the designted fishing e is ppoimtel 0.7 sque ds. pge 5 N P UC7
Setion 7.: Right Tingle Geomet Rememe fom geomet tht simil ight tingles e tingles whee the ute ngles of one tingle e onguent to those of nothe tingle, even if the sides e not the sme length. If the e simil, the tios of the sides of the tingle e the sme. These tios e the sis of tigonomet. If is n ute ngle in the ove ight tingle, then the following tios eist: Tigonometi funtions: Sine: Cosine: Tngent: opposite sin Cosent: hpotenuse os Sent: hpotenuse opposite tn Cotngent: s se hpotenuse opposite hpotenuse ot opposite Emple : Evlute ll si tigonometi funtions of the ngle. Fist of ll, use the Pthgoen Theoem to find the thid side. 5 5 9 sin os tn opposite hpotenuse hpotenuse opposite 5 5 os se ot hpotenuse hpotenuse opposite 5 5 pge N P UC7
Emple : Evlute ll si tigonometi funtions of the ngle. Fist of ll, use the Pthgoen Theoem to find the thid side. 8 08 08 sin os tn opposite hpotenuse hpotenuse opposite hpotenuse s opposite hpotenuse se ot opposite 8 8 8 8 8 8 One thing tht ou m hve notied out osent is tht it is the eipol of sine nd tht sent is the eipol of osine. The sme is tue fo otngent eing the eipol of tngent. In othe wods, fo these thee, ou simpl flip the funtion with whih the e eipols. These e theefoe efeed to s eipol identities. Reipol identities: s sin se os ot tn Also, let s go k to the tingle t the eginning of this setion. We sid tht opposite tn something else. Fo emple, if ou took would hppen:. Howeve, to get, ou ould do sin, think out wht os sin os (Sustitution) sin os (Dividing is multipling the eipol) sin os (Dividing the ) sin os tn (Definition of tngent in tingle) pge 7 N P UC7
sin Theefoe, sine tn, the eipol funtion otngent would e the eipol of tht, os os mening ot. These two identities e known s the quotient identities. sin Quotient identities: sin tn os os ot sin Emple : Use identities to find the et vlue of eh of the fou emining tigonometi funtions of the ute if sin nd os. You know tht sin nd os hve eipol funtions of s nd se, espetivel, so ou n find those tking the eipols. s se sin os 9 To find the tn nd ot, we n use quotient identit nd eipol identit. sin tn os ot tn 9 Theefoe, s, se, tn, nd ot. You led ell us using the Pthgoen theoem t the eginning of this setion. Fo the tingle t the ight, tht would men + =. Tht isn t nthing new, ut n we use the Pthgoen theoem fo othe pplitions? Wht if we wee to divide oth sides (sine wht ou do on one side, ou n do on the othe)? pge 8 N P UC7
, o (Division) (Eponentil popet) Sine ou know fo the tingle tht sin os sin os sin, os, nd, ou n sustitute these in. sin os is onsideed to e fundmentl identit sed on the eltionships we just used. You n use this identit to find othes. Fo emple, wht if ou divided oth sides sin? sin os sin os, o sin sin sin sin sin sin os sin sin sin ot s (Division) (Eponentil popet) (Sustitution) ot s (Sustitution) We now hve nothe fundmentl identit, nmel tht ot s. Wht if we hd divided sin os os insted, though? sin os sin os, o os os os os sin os os os os tn se os (Division) (Eponentil popet) (Sustitution) tn se (Sustitution) Now we hve the lst of the fundmentl identities, tn se. Fundmentl identities: s se ot sin os tn sin os tn ot os sin sin os ot s tn se pge 9 N P UC7
Emple : Use Fundmentl Identities to find the et vlue of eh epession. Do not use lulto. ) tn os sin : tn os sin sin os sin os (Sustitution) sin sin (Cneling out os ) 0 (Suttion) ) sin os : sin os (sin + os ) (Ftoing out ) () (Sustitution sine sin + os = ) (Multiplition) sin 0 ) tn 0 os 0 sin 0 tn 0 os 0 tn 0 tn 0 (Sustitution) 0 (Suttion) Thee e few moe identities we n deive. Note the tingle t the ight. If the ngle is one of the ute ngles in the ight tingle, the othe ute ngle hs to e 90. This mens the following would e tue: sin os 90 os sin 90 tn ot90 s se 90 se s 90 ot tn90 90 This gets us the lst of ou identities whih we will efe to s omplement ngle theoem identities, sine the eltionship is etween two ngles tht would e omplement in the ight tingle. Sine 90 is the sme s dins, we n hve these popeties fo dins s well. Complement Angle Theoem Identities (in degees) (in dins) sin os90 sin os os sin90 os sin tn ot90 tn ot s se90 s se pge 0 N P UC7
se s90 se s ot tn90 ot tn Emple : Use Fundmentl Identities nd/o the Complement Angle Theoem to find the et vlue of eh epession. Do not use lulto. ) os 0 os 50: os 0 os 50 sin (90 0) os 50 (Complement Angle Theoem) sin 50 os 50 (Suttion) (sin 50 + os 50) (Ftoing out ) (Sustituion sine sin + os = ) 0 (Suttion) ) se ot 8: se ot 8 se tn (90 8) (Complement Angle Theoem) se tn (Suttion) (Sine tn + = se, = se tn ) ) sin 0 sin 0 ot 50 ot 50 os 0 os 0 tn 0 ot 50 (Sustitution) tn 0 tn(90 50) (Complement Angle Theoem) tn 0 tn 0 (Suttion) 0 (Suttion) pge N P UC7
Setion 7.: Computing the Vlues of Tigonometi Funtions of Aute Angles Evluting speifi ngle mesues involves tking mentl tip k to when ou hd tken geomet. Let s stt with n isoseles ight tingle. As ou m ell, n isoseles ight tingle is ight tingle whose legs e onguent. Fist of ll, use the Pthgoen Theoem to find the thid side. sin 5 sin os 5 os tn 5 tn opposite hpotenuse hpotenuse opposite hpotenuse s 5 s opposite hpotenuse se 5 se ot 5 ot opposite You hve led developed unit ile. You n lso use the unit ile to get the sme vlues. If ou wee to ete tingle on the unit ile fom the oigin (0, 0) to the point on the unit ile oesponding to n ngle, ou would see the following: Sine ou e going to the left o ight to get the oodinte tht oesponds on the unit ile, tht vlue would e the. The sme is tue fo going up o down to get the oodinte, hene tht vlue eing. Theefoe, getting the tigonometi funtions fo this would e es: sin os tn s se ot Sine = on unit ile (hene the nme UNIT ile), the mth eomes LOT esie. pge N P UC7
pge N P UC7 tn os sin ot se s Unit Cile Tigonomet Funtions: ot se s tn os sin Now look t ou unit ile nd ppl this. 5 ot se 5 s 5 tn 5 os 5 sin 5 You still get the sme vlues s ou did efoe!
Anothe ommon tpe of tingle is the equiltel tingle. If ou split the tingle into hlf fom vete of the tingle to the se of the opposite side, two 0 0 ight tingles e fomed (mening eh ight tingle hs 0 nd 0 ngle. Fo onveniene of lultion, we will let eh side of the equiltel tingle e equl to, mening the split side of the equiltel tingle will hve two segments of length. Use the Pthgoen Theoem to find the thid side. opposite sin 0 sin hpotenuse os 0 os hpotenuse opposite tn 0 tn hpotenuse s 0 s opposite hpotenuse se 0 se ot 0 ot opposite You n do these funtions using unit ile s well. sin0 os0 tn0 s 0 se 0 ot 0 Agin, ou still get the sme vlues s ou did efoe! pge N P UC7
Now, let s ppl these to othe polems ou might enounte. Emple : Find the et vlue of eh epession. Do not use lulto. ) sin 5 tn 0: Using ou unit ile, sin 5 (using the -oodinte of the point t 5) tn0 (using the -oodinte of the point t 5) sin5 tn0 ) os + sin : Using ou unit ile, os (using the -oodinte of the point t 0, o /) sin (using the -oodinte of the point t 5, o /) os sin To evlute tigonometi funtions with lulto, thee e few impotnt things to know: On ou lulto, pess the MODE utton to swith etween dins nd degees. BE SURE TO SET IT IN THE PROPER MODE OR YOU WILL NOT GET THE RIGHT RESULTS. Be sue to use pentheses when neess. Fo emple, putting into ou lulto sin /" is NOT the sme s putting into ou lulto sin (/). (You lulto is ve speifi out using the ode of opetions.) Emple : Use lulto to find the ppopite vlue of eh epession. Round the nswe to two deiml ples. ) sin 7: Pess the MODE utton on ou lulto nd set the mode to degees. Tpe in sin(7). You should get vlue of 0.7. ) os : 7 Pess the MODE utton on ou lulto nd set the mode to dins. Tpe in os(/7). You should get vlue of 0.. pge 5 N P UC7
Emple : Solve fo in the pitue elow. Emple : Solve fo nd in the pitue elow. To solve this, use wht ou know to set up tigonometi tio. In this se, the est tio to use is the sine funtion, speifill the tio whee 0 sin To get to, ou must use the invese sine funtion. In othe wods, 0 sin 0.878 dins o 50. Agin, use the tigonometi tios ou n setup with the given infomtion so tht the onl vile is the one fo whih ou e solving. tn 0 tn 0 os 0 os 0 os 0 () Thee e el-life pplitions fo the use of tigonometi funtions. Emple 5: You wnt to mesue the width oss lke efoe ou swim oss it. To mesue the width, ou plnt stke on one side of the lke, dietl oss fom the dok. You then wlk 5 metes to the ight of the dok nd mesue 5 ngle etween the stke nd the dok. Wht is the width w of the lke? Fist, dw pitue to undestnd the polem. As ou hve done efoe, set up tigonometi tion sed on wht ou know. In this se, the tngent funtion would e the est seletion. w tn 5 w 5 tn 5 5 5 5 m Thee e two speifi tpes of ngles used in el-life pplition onening tigonomet. One is the ngle of elevtion, the ute ngle fomed the line of sight upwd nd the hoizontl, mening the ngle one looks up t fom the gound to see something. The ngle of depession is the opposite; it is the ute ngle fomed the line of sight downwd nd the hoizontl, mening the ngle one looks down fom line pllel to the gound to see something. pge N P UC7
Emple : In the digm shown elow, kite t point C is eing held sting fom point A tht is 0 ft long. The ngle of elevtion of the kite fom point A is 0. An oseve t point B is 0 ft fom the point on the gound D dietl elow the kite. Wht is the ngle of elevtion of the kite fom point B. In this se, ou e looking fo ngle B. Sine ou do not know the vlue of CD o of BC, ou must fist find one of them. The onl one tht n e esil found is CD setting up tigonometi tio. CD sin 0 CD 0 sin 0 0(0.) 0.5 ft 0 Now tht ou know the vlue of CD, ou m use tigonometi tio to find the ngle B. t n B CD BD tn B 0.5 0 B tn 0.5.0 0 pge 7 N P UC7
Setion 7.: Tigonometi Funtions of Genel Angles Rell fom the pevious setion tht we detemined the following: If ou hve ile with point (, ) on the teminl side of n ngle, the tigonometi funtions fo this would e es: sin os tn s se ot Emple : The point, is on the teminl side of n ngle. Find the et vlue of eh of the si tigonometi funtions of. Sine nd theoem., ou n find the thid side (the hpotenuse) using the Pthgoen + = (Pthgoen theoem, djusted fo pitue ove) (Sustitution) (Eponentition) 9 (Addition) 9 (Sque oot) Now tht ou know wht is, ou n figue out ll of the tigonometi funtions. sin os pge 8 N P UC7
tn s se ot 8 9 Wht if point on teminl side of n ngle does not fom tingle? This would e the se if suh point fell on the - o -is. Emple : Find the et vlues of eh of the si tigonometi funtions of. would men = 90, so the point (0, ) would e on the teminl side of tht ngle. Sine ou e going up fom (0, 0) to get to tht point, the =. = 0, nd =, so ou n use wht ou hve efoe to figue out the tigonometi funtion vlues. 0 sin os 0 tn Undefined 0 0 s se Undefined ot 0 0 Onl t the - o -is n ou potentill hve undefined vlues fo ou tigonometi funtions. Going k to the fomuls ou just used, notie tht lws hs to e positive given it is positive distne fom the oigin (0, 0), espeill sine ou n t hve negtive dius. Howeve, nd n e positive o negtive. Sine sin nd s involve, is positive in qudnts I nd II nd negtive in qudnts III nd IV, mening sin nd s e positive in qudnts I nd II nd negtive in qudnts III nd IV. Simill, os nd se involve, nd is positive in qudnts I nd IV nd negtive in qudnts II nd III, mening sin nd s e positive in qudnts I nd IV nd negtive in qudnts II nd III. Finll, tn nd ot involve oth nd. Fo tn nd ot to e positive, nd would oth hve to e positive o oth hve to e negtive (so tht, when the divide, the e positive). The onl qudnts in whih these ou e in qudnts I nd III. This mens tht tn nd ot ould onl e negtive in qudnts II nd IV. pge 9 N P UC7
Whee e tigonometi vlues positive o negtive? Qudnt whee is sin nd s os nd se tn nd ot I + + + II + III + IV + Emple : Nme the qudnt in whih the ngle lies if sin < 0 nd os > 0. sin < 0, o sin is negtive, in qudnts III nd IV. os > 0, o os is positive, in qudnts I nd IV. Theefoe, sin < 0 nd os > 0 onl in qudnt IV. Bk in the fist setion of this hpte, we disoveed tht ou n hve oteminl ngles, ngles tht hve the sme teminl side. Fo us, it tends to e esie if the e positive (whih e onsideed to e in stndd position) thn if the e negtive. In the fist setion, to hnge those ngles into positive ngles, we sid ou ould just dd full ile, o 0. Sine we know full ile in dins is, ou ould dd if the ngle is in dins. This is lso tue if the ngle is igge thn 0 o dins, mening ou ould just sutt full ile, o 0 o dins. Emple : Find the smllest positive ngle of eh ngle. ) 90: Sine this is igge thn ile, simpl sutt full ile, o 0, fom it. 90 0 = 0 ) 0: Sine this is smlle thn ile, simpl dd full ile, o 0, to it. 0 + 0 = 0 ) 9 : Sine this is igge thn ile, simpl sutt full ile, o, fom it. 9 9 8 Notie tht we got nie, smll ngles in ) nd ) tht e in ou fist qudnt of the unit ile we peviousl eted (nd fo whih ou hve een hopefull studing). These e lso ngles tht e etween 0 nd 90 o etween 0 dins nd dins. These kinds of ngles e efeene ngles. The e positive ute ngles fomed the oteminl side of nothe ngle. The following polem will demonstte this: Emple 5: Find the efeene ngle of eh ngle: ) 5, ) f) 7. 7 7, ) 00, d) 0, e), These e ll dwn in the following digm: pge 0 N P UC7
Rememe tht, to e efeene ngle, eh ngle must e positive ute ngle oteminl to the side of the ngle. These e eh leled ove. ) 5: is etween 5 nd 80. To find, ou just hve to sutt the two ngles. In othe wods, = 80 5 = 5. Theefoe, the efeene ngle fo 5 is 5. ) 7 : 7 is etween nd. To find, ou just hve to sutt the two ngles. In othe 7 7 pge N P UC7 7 is. wods, =. Theefoe, the efeene ngle fo ) 00: is etween 00 nd 0, whih is the sme s 0. To find, ou just hve to sutt the two ngles. In othe wods, = 0 00 = 0. Theefoe, the efeene ngle fo 00 is 0. d) 0: Rell tht if ou dd full ile, s we did efoe, ou get positive ngle. 0 0 = 0, mening 0 is the sme s 0, nd is etween 0 nd 80. To find, ou just hve to sutt the two ngles. In othe wods, = 0 80 = 0. Theefoe, the efeene ngle fo 0 is 0. e) f) 7 : Rell tht if ou dd full ile, s we did efoe, ou get positive ngle. 7 7 5, mening 5 7 is the sme s, nd is etween nd. To find, ou just hve to sutt the two ngles. In othe wods, 5 5 7 θ. Theefoe, the efeene ngle fo is. 7 : Rell tht if ou dd full ile, s we did efoe, ou get positive ngle. 7 7 8 7, mening is the sme s, nd is etween nd 0. To find, ou just hve to sutt the two ngles. In othe wods, θ 0. 7 Theefoe, the efeene ngle fo is. 5
The nie thing out using efeene ngles is tht ou don t hve to memoize the entie unit ile, ell just the fist qudnt, if ou know how to ppl tigonometi funtions sed on wht qudnt ou e in. Refeene ngles hve the sme tigonometi vlues s the oiginl ngles, ut ou hve to detemine if the e negtive o positive sed on wht qudnt ou e in. Emple : Use the efeene ngle to find the et vlue of eh epession without using 7 lulto: ) sin 5, ) os 7, ) tn 00, d) s 0, e) se, 7 f) ot. ) sin 5: We peviousl detemined the efeene ngle fo 5 is 5. On the unit ile, the point on the oteminl side of this ngle is,. Sine sin, nd 5 is whee is positive, the sine will e positive. Theefoe, ) 7 os : sin. We peviousl detemined tht the efeene ngle fo 7 is. On the unit ile, the point on the oteminl side of this ngle is,. Sine os, 7 nd is whee is negtive, the osine will e negtive. Theefoe, os. ) tn 00: We peviousl detemined tht the efeene ngle fo 00 is 0. On the unit ile, the point on the oteminl side of this ngle is,. Sine tn, nd 00 is whee is positive nd is negtive, the tngent will e negtive. Theefoe, tn. d) s 0: We peviousl detemined tht the efeene ngle fo 0 is 0. On the unit ile, the point on the oteminl side of this ngle is,. Sine s, nd 0 is whee is negtive, the osent will e negtive. Theefoe, s. pge N P UC7
7 7 e) se : We peviousl detemined tht the efeene ngle fo is. On the unit ile, the point on the oteminl side of this ngle is,. Sine se, 7 nd is whee is negtive, the sent will e negtive. Theefoe, se. 7 7 f) ot : We peviousl detemined tht the efeene ngle fo is. On the unit ile, the point on the oteminl side of this ngle is,. Sine ot, nd is whee is positive nd is positive, the otngent will e positive. Theefoe, ot. You n ppl the ide of onsideing negtives nd positives to ngles tht e not efeene ngles. Emple 7: Find the et vlue of eh of the emining tigonometi funtions of if nd 80 < < 70. os Sine ou know this is in the thid qudnt, ou know tht nd will oth e negtive, so dw pitue to show this. Sine os, = nd =. You know will e negtive sine it is in the thid qudnt (nd going down). To figue it out, simpl use the Pthgoen Theoem. 9 8 8 Sine ou now know tht = (negtive euse of eing in the thid qudnt), ou hve evething ou need to figue out the emining tigonometi identities. sin s pge N P UC7
os se tn ot pge N P UC7