Section 2.2 Trigonometry: The Triangle Identities

Transcription

1 Se. 2.2 Trigonometry: The Tringle Identities 7 Setion 2.2 Trigonometry: The Tringle Identities DJENT ND OPPOSITE SIDES The study of tringle trigonometry is entered round the ute ngles in right tringle. In right tringle, we ll the leg tht forms n ute ngle with the hypotenuse the djent side, or side djent, to tht ngle; the other leg is the opposite side, or side opposite. For exmple, in the tringles elow, side is djent to nd side is opposite. lso, in referene to ngle, side is djent to nd side is opposite. Side opposite ngle Side djent to ngle Side djent to ngle Side opposite ngle THE SINE ND OSINE FUNTIONS In funtion, f(x), the vlue in prentheses is lled the rgument of the funtion. For exmple, if f(x) = 2x, then x is the rgument. If we reple x with nd get f() = 2(), then is the rgument. In tringle trigonometry, sine nd osine re funtions, nd the rguments re the ute ngle mesures in right tringle*. For exmple, in the sine of 70, written sin(70 ), 70 is the rgument; likewise, in the osine of, written os(), the rgument is. In right tringle : The sine of n ute ngle is the rtio of the side opposite the ngle to the hypotenuse. s rtio, sin() = the side opposite the hypotenuse Hypotenuse Side opposite ngle = opp hyp the side opposite sin() = the hypotenuse Hypotenuse Side opposite ngle = opp hyp *s with ny funtion, trigonometri funtions must hve n rgument; e.g., os without n rgument is meningless. Se. 2.2 Trigonometry: The Tringle Identities 7 Roert H. Prior, 2016 opying is prohiited.

2 8 Se. 2.2 Trigonometry: The Tringle Identities The osine of n ute ngle is the rtio of the side djent to the ngle to the hypotenuse. s rtio, os() = the side djent the hypotenuse = dj hyp the side djent os() = the hypotenuse = dj hyp Hypotenuse Side djent ngle Hypotenuse Side djent ngle To this point we hve the following: sin() = opp hyp = sin() = opp hyp = os() = dj hyp = os() = dj hyp = Exmple 1: Given eh tringle, find the sine nd osine of ngles nd. Rtionlize the denomintor if neessry. ) in in in sin() = in in = os() = in in = sin() = in in = os() = in in = Se. 2.2 Trigonometry: The Tringle Identities 8 Roert H. Prior, 2016 opying is prohiited.

3 Se. 2.2 Trigonometry: The Tringle Identities ) sin() = m m = = = m m os() = m m = 1 = 1 = m sin() = m m = 1 = os() = m m = = Importnt note: In the exmples ove, the rgument of eh funtion is n ngle ( or ), ut wht is the result (output) of the funtion? Eh result is rtio in whih the units of mesure divide out. So, eh output is rel numer independent of ny units. From this point on, the side mesures of eh tringle will not inlude units. THE TNGENT FUNTION third trigonometri funtion, the tngent, is rtio ompring the two legs of the tringle. s rtio, tn() = the side opposite the side djent = opp dj tn() = opp dj = tn() = opp dj = Se. 2.2 Trigonometry: The Tringle Identities Roert H. Prior, 2016 opying is prohiited.

4 0 Se. 2.2 Trigonometry: The Tringle Identities Exmple 2: Given the right tringle elow, find the tngent of ngles nd. ) ) nswer: ) tn() = ) tn() = = tn() = tn() = = 1 It is importnt to memorize the sine, osine, nd tngent rtios. SOH-H-TO helpful id in rememering the tringle trig reltionships is Soh-h-To, whih mens: Soh: Sine is Opposite Hypotenuse h: osine is djent Hypotenuse Opposite To: Tngent is djent The Reiprol Funtions Eh of the three mjor trig funtions hs reiprol. They re listed elow. Note: 1 mens the reiprol of. i.e., the reiprol of n e represented y 1. THE SENT FUNTION The sent of n ngle,, revited se(), is the reiprol of the osine funtion: se() = 1 os() nd os() = 1 se() Memorize these! For exmple, if os() = 2 7, then se() = 7 2 ; lso, if se() =, then os() = 1. Se. 2.2 Trigonometry: The Tringle Identities 0 Roert H. Prior, 2016 opying is prohiited.

5 Se. 2.2 Trigonometry: The Tringle Identities 1 We n lso relte the sent funtion to the right tringle: se() = hypotenuse the side djent = hyp dj However, it is ommon to find se() only fter first using the tringle to find os(). In other words, we rrely use the right tringle to find se(). os() = dj hyp =, so se() = os() = dj hyp =, so se() = Exmple : Given the tringle elow, find the sent of ngles nd. Rtionlize the denomintor if neessry. ) ) nswer: For eh, first find the osine of the ngle, nd then find the sent s the reiprol of the osine. In prt ), we deide whether to rtionlize the denomintor only fter we see whether the sent funtion requires it. ) os() =, so se() = ; os() =, so se() = ) os() = = 1 (don t rtionlize yet) ; se() = 1 = os() = = (don t rtionlize yet) ; se() = Se. 2.2 Trigonometry: The Tringle Identities 1 Roert H. Prior, 2016 opying is prohiited.

6 2 Se. 2.2 Trigonometry: The Tringle Identities THE OSENT FUNTION The osent of n ngle,, revited s(), is the reiprol of the sine funtion: s() = 1 sin() nd sin() = 1 s() Memorize these! s with the sent funtion, it is ommon to find se() only fter first using the tringle to find sin(). In other words, we rrely use the right tringle to find s(). sin() = opp hyp =, so s() = sin() = opp hyp =, so s() = Exmple : Given the tringle elow, find the osent of ngles nd. Rtionlize the denomintor if neessry. ) ) nswer: For eh, first find the sine of the ngle, nd then find the osent s the reiprol of the sine. In prt ), we deide whether to rtionlize the denomintor only fter we see whether the osent funtion requires it. ) sin() =, so s() = ; sin() =, so s() = ) sin() = = (don t rtionlize yet) ; s() = sin() = = 1 (don t rtionlize yet) ; s() = 1 = Se. 2.2 Trigonometry: The Tringle Identities 2 Roert H. Prior, 2016 opying is prohiited.

7 Se. 2.2 Trigonometry: The Tringle Identities THE OTNGENT FUNTION The otngent of n ngle, revited ot(), is the reiprol of the tngent funtion: ot() = 1 tn() nd tn() = 1 ot() Memorize these! Relting the otngent funtion to the right tringle, we get: ot() = the side djent the side opposite = dj opp Note: In tringle trigonometry, it is ommon to first find the tngent of n ngle efore finding the otngent, ut it is lso ommon to know the tringle reltionship defined ove. ot() = dj opp = ot() = dj opp = tn() = nd tn() = Exmple : Given the tringle elow, find the otngent of ngles nd. Rtionlize the denomintor if neessry. ) ) nswer: ) ot() = ) ot() = = 1 ot() = ot() = = Se. 2.2 Trigonometry: The Tringle Identities Roert H. Prior, 2016 opying is prohiited.

8 Se. 2.2 Trigonometry: The Tringle Identities O-FUNTIONS You ve proly notied some interesting onnetions etween some of the trigonometri funtions. For exmple, every funtion hs o-funtion: sine nd osine re o-funtions tngent nd otngent re o-funtions sent nd osent re o-funtions Tht s proly not surprise. Wht might e surprise, though, is tht sent is the reiprol of osine, wheres osent is the reiprol of sine. Plese lern this well memorize these reltionships s they re very importnt to the suess of lerning trigonometry. The onnetion etween pir of o-funtions is omplementry ngles. Speifilly, the sine of n ngle is the osine of its omplement: sin() = os(0 ) For exmple, 20 nd 70 re omplementry ngles so it must e tht sin(20 ) = os(70 ) tn(20 ) = ot(70 ) se(20 ) = s(70 ) nd nd nd sin(70 ) = os(20 ) tn(70 ) = ot(20 ) se(70 ) = s(20 ) THE UTE NGLES IN RIGHT TRINGLE right tringle hs two ute ngles tht re omplementry to eh other, nd the o-funtion reltionships n e seen using ny right tringle: sin() = = os() tn() = = ot() se() = = s() nd nd nd sin() = = os() tn() = = ot() se() = = s() These o-funtion reltionships re useful when finding the six trigonometri funtion vlues for the two ute ngles in tringle. For exmple, one you know tht tn() =, then you lso know tht, without ny work, ot() =. Se. 2.2 Trigonometry: The Tringle Identities Roert H. Prior, 2016 opying is prohiited.

9 Se. 2.2 Trigonometry: The Tringle Identities Setion 2.2 Fous Exerises For eh tringle, find ll six trig funtion vlues for eh ute ngle. e sure to simplify eh ompletely, inluding rtionlizing ny denomintors tht need it sin() = sin() = os() = os() = tn() = tn() = ot() = ot() = se() = se() = s() = s() = sin() = sin() = os() = os() = tn() = tn() = ot() = ot() = se() = se() = s() = s() = Se. 2.2 Trigonometry: The Tringle Identities Roert H. Prior, 2016 opying is prohiited.

10 6 Se. 2.2 Trigonometry: The Tringle Identities sin() = sin() = os() = os() = tn() = tn() = ot() = ot() = se() = se() = s() = s() = sin() = sin() = os() = os() = tn() = tn() = ot() = ot() = se() = se() = s() = s() = Se. 2.2 Trigonometry: The Tringle Identities 6 Roert H. Prior, 2016 opying is prohiited.