Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that the equation is meaningless if eithe side becomes undefined fo some value. Fo eample, take the following identit: This is an identit because is the same as and the + facto divides out. Howeve, if ou substitute in into the equation, one side is undefined, meaning that the domain of both sides is (, ) (, ) fo the identit. This domain is called the domain of validit ce this domain allows values that do not make the identit invalid o undefined. Back in Chapte 7, ou leaned that each basic tigonometic function has a ecipocal function. Fo eample: cot This means taking the ecipocal of each ecipocal function will etun to the oiginal tigonometic function. In othe wods, cot Also, fom Chapte 7, ou leaned that, based on the unit cicle as well as efeence tiangles, a point (, ) on the teminal side of a tiangle with disce fom the oigin can eveal the following tigonometic functions and calculations fo the following efeence tiangle: cot Fom this, we can deive othe identities that ae commonl used. cot page N P UC8S-6
Fo the tiangle shown peviousl, the Pthagoean theoem eveals that +. You can use this identit to pove othe identities that ae commonl used. Fo eample, Pthagoean Identit #: Anothe Pthagoean identit comes ight fom this one: Pthagoean Identit #: You can also use the fist Pthagoean identit to find anothe Pthagoean identit. cot cot Pthagoean Identit #: cot Thee ae also what ae called cofunction identities, meaning identities deived fom the complements of angles. Realize that each acute angle is a complement of the othe acute angle of a ight tiangle because if one angle is the ight angle of the tiangle, and the sum of the angles of an tiangle is 80, then the acute angles must add up to 90. page N P UC8S-6
Fo the tiangle above, ealize, ug the tigonometic functions of befoe, that the A 90 C, o A C. A A A C C C A A cot A C C cot C Notice the following: A C A C A cot C A C A C cot A C Realizing again that A C, the following cofunction identities eist fo : Cofunction Identities: cot cot Back in Chapte 7, ou leaned that e, coe, and gent functions wee identified as odd o even functions. B definition, a function is odd if f( ) f() and is also smmetic about the oigin. A function is even if f( ) f() and is also smmetic about the -ais. Since e and gent functions ae odd functions and the coe function is an even function, the following identities eist: ( ) ( ) Odd Identities ( ) cot( ) cot Even Identities ( ) ( ) One can use the pevious identities to simplif moe comple epessions. page N P UC8S-6
Eample: Use basic identities to simplif the epession. #: : #: (/ ): (/ ) #: : Eample: Simplif the epession to eithe o. #: ( ): ( ) ( ) #5: cot ( ) cot (/ ): cot ( ) cot (/ ) cot Eample: Simplif the epession to eithe a const o a basic tigonometic function. #6: / : / cot cot cot #7: ( + ) ( + cot ): ( + ) ( + cot ) + cot ( ) + ( cot ) + One can also simplif b factoing, epanding, combining factions, and ug identities. Eample: Use the basic identities to change the epession to one involving onl es and coes. Then simplif to a basic tigonometic function. #8: ( )( + cot ): cot cot page N P UC8S-6
#9: : Factoing is often useful fo solving equations. Nomall, in ode to facto an epession, ou need the tems in the same tigonometic fom; in othe wods, all tems need to be consts o need to be the same tigonometic function. Much in the same wa ou can facto + + into ( + )( + ), ou can facto something like + + into ( + )( + ) b the same methods ou leaned to facto befoe. Eample: Wite each epession in factoed fom as an algebaic epession of a gle tigonometic function. #0: + : Since all tigonometic functions have to be the same, it might be helpful to change one of the tems into the fom of anothe tem. is the same as, so change that fist. #: : cot cot Sometimes the et to simplifing is getting id of the denominato. As ou did with adicals and with imagina numbes in the denominato, ou ma need to multipl b the conjugate of the denominato and simplif. Eample: Wite each epession as an algebaic epession of a gle tigonometic function. #: : Poving tigonometic identities is much like ou have been doing in simplifing epessions. It involves making both sides equal to each othe. In this tion, howeve, ou must ewite one o both sides to look the same as the othe. Thee ae some helpful hints to get thee: page 5 N P UC8S-6
Stat with the moe complicated side and wok towad getting a esult of the less complicated side. T conveting evething to e and coe epessions when all else fails. Combine factions b combining them ove a common denominato. Use the algebaic identit (a + b)(a b) a + b to get a Pthagoean identit if it appeas that ou might be able to get one. Alwas set up some goal to get and manipulate epessions of one o both sides to get that. Realize one thing when poving tigonometic identities: eve intemediate step ou do should povide an epession that is equivalent to the fist. Eample: Pove the identit. #: ( )( + cot ) + In this case, the left side is the moe complicated epession, so stat thee. #: ( )( + cot ) + + cot + + + u u u u In this case, the left side is the moe complicated epession, so stat thee. u u u u u u u u u u u u u u u u u #5: In this case, the left side is the moe complicated epession, so stat thee. #6: Yet again, the left side is the moe complicated epession, so stat thee. page 6 N P UC8S-6
Up to this point, ou have onl had to simplif one side to pove the identit. In the net poofs, ou ma have to change both sides. Eample: Pove the identit. #7: T multipling the left side s denominato b its conjugate to t to simplif things. #8: To get in the denominato on the ight, it would impl having to multipl and togethe to get a middle tem. Theefoe, t it. cot #9: cot Since this appeas almost impossible, t conveting evething to e and coe and see whee it takes ou. page 7 N P UC8S-6
page 8 N P UC8S-6 cot cot Recall that a b (a b)(a + ab + b ), so ( )( + + ). #0:
Section 8.5: Sum and Diffeence Fomulas Conta to what man people believe, just because things ae being added in an opeation does not mean the can alwas be sepaated into two diffeent opeations. Fo eample, man students believe that o that ( + ) +. NEITHER OF THOSE ARE TRUE!!!!!!!!!!!!!!! In the same wa, ou cannot sepaate a e o coe o gent if addition is going on. Rathe than list the poof hee, the poofs fo sum and diffeence identities of es and coes is on page 69 in ou online tetbook, and these will be discussed in class. Howeve, the identities fo establishing sum and diffeence identities is as follows: u v u v u v u v u v u v Coes of Sums and Diffeences: u v u v u v u v u v u v u v u v u v Sines of Sums and Diffeences: u v u v u v Ug the sepaate fomulas fo sums and diffeences of es and coes, one can deive the fomula fo the sum and diffeences of gents. u u v u v v u v Tangents of Sums and Diffeences: u v u v u v u v u v In othe wods, ou must use one of these fomulas to pevent something along the lines of. page 9 N P UC8S-6
The idea behind ug these sum and diffeence identities is to use eact values ou know fom efeence tiangles athe than appoimations fom ou calculato to epess the e, coe, and gent values of non-efeence-tiangle angles. Eample: Use a sum o diffeence identit to find an eact value. #: 5: 5 is the esult of 5 0, both angles of which ae efeence angles fo which ou should know the eact e and coe values. u v u v u v 5 5 0 5 6 0 5 0 6 #: : is the esult of, which is the same as, both angles of which ae efeence angles fo which ou should know the eact e and coe values. u v u v u v 6 6 5 5 #: : is the esult of +, which is the same as +, both angles of which 6 ae efeence angles fo which ou should know the eact gent values. u v u v u v 5 6 6 6 9 6 6 9 6 Eample: Wite the epession as the e, coe, o gent of an angle. #: : This epession matches up with that of the e of a sum. In othe 5 5 wods, 5 7 5 5 5 0 0 0 page 0 N P UC8S-6
#5: : 7 7 This epession matches up with that of the coe of a diffeence. In othe wods, 7 7 7 One can also use these sum and diffeence identities to pove othe identities. Eample: Pove the identit. #6: - : - #7: : ( + ) ( + ) ( + ) ( ) ( + ) ( ) ( ) Section 8.6: Double-angle and Half-angle Fomulas In Section 8.5, ou leaned how to find the e, coe, and gent values fo angles that esult fom sums and diffeences. Thee ae also fomulas o identities that esult fom when u v, esulting in what ae called double-angle identities. These identities can be found fo each tigonometic function. Double-Angle Identit fo Sine: u (u + u) u u + u u u u + u u u u Double-Angle Identit fo Sine: Double-Angle Identities fo Coe: u (u + u) u u u u u u u u u page N P UC8S-6
Fist Double-Angle Identit fo Coe: u u u The double-angle identit fo coe can also be epessed entiel in tems of coes and consts. u u u u ( u) u + u u Second Double-Angle Identit fo Coe: u u The double-angle identit fo coe can also be epessed entiel in tems of es and consts. u u ( u) u u Thid Double-Angle Identit fo Coe: Double-Angle Identit fo Tangent: u (u + u) u u u u u u Double-Angle Identit fo Tangent: u u u u u Sometimes it helps to educe powes fom u, u, and u to emove the squae, paticulal in applications of calculus. Powe-Reducing Identit fo Sine: Powe-Reducing Identit fo Sine: Poof fo Powe-Reducing Identit fo Sine: u u page N P UC8S-6
u u u u u u Powe-Reducing Identit fo Coe: Powe-Reducing Identit fo Coe: Poof fo Powe-Reducing Identit fo Coe: u u u u u u Powe-Reducing Identit fo Tangent: Powe-Reducing Identit fo Tangent: u u u u u Poof fo Powe-Reducing Identit fo Tangent: u u u u u u u page N P UC8S-6
u u u u u u Related to the powe-educing identities ae the half-angle identities, useful when ou ae calculating halves of efeence tiangles o othe calculated angles. NOTE: Thee will not be much focus on that in this tion. The following ae the half-angle identities: u u Eample: Pove the identit. u u u u u u u u u #: : ( + ) + + #: : page N P UC8S-6
#: 8 : () ( ) ( ( + )) ( + ) ( + ) ( ) ( ) 8 8 8 : ( ) #: 8 8 8 Half-angle identities can be used to find eact values without the use of a calculato. Eample: Use half-angle identities to find an eact value without a calculato. #5: 5: 5 is half of 0. 0 5 (positive ce in the fist quadant) 0 page 5 N P UC8S-6
page 6 N P UC8S-6 #6: 7/: 7/ is half of 7/6. 6 7 6 7 7