MAT 75: Intrdutin t Mthemtil Anlysis Dr. A. Rzenlyum Trignmetri Funtins fr Aute Angles Definitin f six trignmetri funtins Cnsider the fllwing girffe prlem: A girffe s shdw is 8 meters. Hw tll is the girffe if the sun is 8 t the hrizn? Trignmetri funtins tht we intrdue here, llw t slve this nd mny mre prlems tht invlve ngles nd sides f tringles. We will slve the ve prlem lter in this setin. T pprh suh prlems, let s strt with definitin f trignmetri funtins fr ute ngles. Cnsider n ute ngle θ : θ Trignmetri funtins (in shrt trig funtins) tke this ngle s its rgument (s input) nd ssign sme numeril vlues t it (utput vlues). Yu will see shrtly wht extly these vlues re. Beuse ngle θ is ute, we n lwys nstrut right tringle with this ngle: θ By prprtinlity prperties f similr tringles, the rtis f sides f this tringle d nt depend n the size f the tringle; insted, they depend n the ngle θ nly. In ther wrds, if we tke tw right tringles with the sme ngle θ, ut different sizes, then the rtis f sides remil the sme. Trignmetri funtins re extly these rtis. It is esy t see tht there is ttl f six psile rtis f the sides in tringle. Here re ll f them: /, /, /, /, /, /. S, there re six trignmetri funtins. Eh f them hs its wn nme nd nttin. The fllwing tle defines ll six trig funtins fr ngle θ.
Funtin Nme Funtin Nttin Definitin sine sine tngent tngent sent sent sin θ s θ tn θ t θ se θ s θ It my seem tht it is diffiult t memrize ll f these funtins. A simple dvie (ut, perhps, nt s simple t fllw) is just t memrize them s yu wuld the multiplitin tle. Frm the ve six trig funtins, the fllwing three re the mst requently used: sine, sine, nd tngent. They re lled si trig funtins. The ther three re reiprls t sis: tngent is reiprl t tngent, sent is reiprl t sine, nd sent is reiprl t sine: t θ =, se θ =, s θ =. tnθ sθ sinθ Sme peple like the fllwing mnemni devie ShChT t memrize the definitin f si trig funtins. It wrks like this. In the ve right tringle, we n tret legs nd s ppsite nd djent t the ngle θ : Hyptenuse θ Adjent Nw, the definitin f sine, sine, nd tngent n e refrmulted s Oppsite sin θ = Oppsite/Hyptenuse s θ = Adjent/Hyptenuse tn θ = Oppsite/Adjent The first three letters f the wrd ShChT men: Sine is the rti f Oppsite leg t Hyptenuse, nd s n.
Trig Funtins fr Speil Angles We hve intrdued speil ngles 0, 45 nd 60 in presius setin s ngles in speil right tringles 0 60 nd 45 45. Here we lulte si trig funtins sine, sine nd tngent fr these ngles. Beuse trig funtins d nt depend n the size f tringle, fr lultins, we n hse ny vlue fr ne f the sides. Let s selet the vlue f fr the shrtest leg fr 0 60 tringle nd fr th legs fr 45 45 tringle. Rell tht in 0 60 tringle, hyptenuse is twie s the shrtest leg (this leg is ppsite t 0 ngle), s the hyptenuse is. Then y Pythgren Therem the ther leg is 45 45 tringle, hyptenuse is pitures 0 Nw, we use the definitin f si trig funtins. =. Fr + =. We n drw the fllwing tw 0 ngle: ppsite side is, djent side is, nd hyptenuse is. Therefre, sin 0 =, s0 =, tn 0 = =. 60 ngle: ppsite side is, djent side is, nd hyptenuse is. Therefre, sin 60 =, s60 =, tn 60 =. 45 ngle: ppsite side is, djent side is, nd hyptenuse is. Therefre, sin 45 = =, s 45 = =, tn 45 =. Let s summrize these results in the fllwing tle 60 45 45 Angle θ 0 sin θ s θ tn θ 45 60
4 If it is diffiult t memrize this tle, try t reprdue the ve tw pitures nd use them t get vlues f trig funtins y their definitins. Als, there is simple pttern fr the sine f speil ngles: it is frmul n /. Just put n =,, in it: n θ 0 45 60 sin θ = n / / / / Wrking with ritrry ute ngles If ngles re nt speil, t find the vlues f si trig funtins, we n use ttns sin, s nd tn n sientifi r grphing lultr. Exmple. Let s slve the girffe prlem, stted t the eginning f this setin. We n drw rrespnding piture like this 8 Shdw = 8 m girffe =? Fr the 8 ngle, girfffe is the ppsite side, nd shdw djent. A suitle trig funtin is tngent (rtin f the ppsite side t djent). Let s dente girffe s shdw g y s nd girffe s height y g. We hve tn 8 =. Frm here, s g = stn 8 = 8 0.57 = 4.5 m. Exmple. Nik lunhed kite n 0-m thred. The thred frms 7 ngle t the hrizn. At wht ltitude is the kite flying? Slutin. Here is the rrespnding piture t 7 h
5 Let s dente the length f the thred y t. This is the hyptenus nd t = 0. The prlem is t find height h whih is the ppsite side t the 7 ngle. A suitle trig funtin is h sine (rtin f the ppsite side t hyptenuse). We hve sin 7 =. Frm here, t h= t sin 7 = 0 0.608 = 7. m. Exmple. A ldder is lening ginst wll t the 56 ngle t the hrizn. Wht is the length f the ldder if its lwer end is m frm the wll? Slutin. Here is the rrespnding piture wll ldder 56 d = m Let s dente the ldder s length y l nd the distne frm its lwer end t the wll y d. We hve d = m. This is djent side fr the 56 ngle. The prlem is t find l. This is hyptenuse. A suitle trig funtin is sine (rtin f the djent side t hyptenuse). We hve s56 = d l. Frm here, d l = = =.5 m. s 56 0.57 Trig funtins llw ls t find ngles in right tringles when inf ut sides is knwn. T slve suh prlems, first identify, similr t previus exmples, whih trig funtin reltes t given prlem nd find the vlue f this funtin. Then yu n find the ngle y using uttns sin, s nd tn n lulr. These uttns lulte the vlues f s-lled inverse trignmetri funtins. These funtins restre n ngle frm the vlues f rrespnding trig funtins. Exmple 4. Cnsider right tringle B A C Use the fllwing infrmtin t find ngle A. ) =, =. ) =, =. ) =, =.
6 Slutin. ) Here suitle trig funtin is sine (rti f ppsite side t hyptenuse): sin A = =. Using lultr, A = sin = 4 ) Here suitle trig funtin is sine (rti f djent side t hyptenuse): s A = =. Using lultr, A = s = 48 ) Here suitle trig funtin is tngent (rti f ppsite side t djent): tn A = =. Using lultr, A = tn = 4