Trigonometric functions - Mathcentre

x 0 ◦ 45 ◦ 90 ◦ 135 ◦ 180 ◦ 225 ◦ 270 ◦ 315 ◦ 360 ◦sin x 0 0.71 1 0.71 0 −0.71 −1 −0.71 0cosx 1 0.71 0 −0.71 −1 −0.71 0 0.71 1tanx 0 1 * −1 0 1 * −1 0Youcanseefromthetablethattherearesomevaluesof xforwhich cosx = 0,andso tanxisnotdefinedforthesevaluesof x. Theseareat 90 ◦ , 270 ◦ ,andalsoothervaluesdifferingfromthem‘bymultiplesof 180 ◦ .f(x)f(x) = tan x2−360°10−1360° 720°x−2Noticethat,whenwegetto x = 360 ◦ ,thegraphsof sin xand cosxrepeatthemselves.As tan xdependsononly sin xand cosx,thegraphof tanxmustalsorepeatitself. But tanxrepeatsitselfmoreoftenthan sin xand cosx. Itrepeatsitselfevery 180 ◦ .So tanxhasaperiodicityof180 ◦ ,or πifyouworkinradians.Noticealsothat,unlike sin xand cosx,thefunction tan xdoesnothavetoliebetween −1and1.Infact tanxcantakeanyvalue.Wecanalsodefineaninversetangentfunction,andtodothiswemustrestrictthedomainof f(x) = tan x. Infactweneedtothinktwiceashardthistimebecause,aswellasmakingsurethatwegetonlyoneanglegivingeachtangentvalue,wemustalsoavoidtryingtodefinef(x) = tan xoveraregionwherethereisazerocosinevalue.Sowecannotdefine f(x) = tan xfor 0 ◦ ≤ x ≤ 180 ◦ becauseat x = 90 ◦ thereisapointwhere f(x)isnotdefined.Wegetaroundthisbyinsteadrestrictingourdomainto −90 ◦ < x < 90 ◦ ,excludingthevalues −90 ◦ and 90 ◦themselves. Thisdomaingivesoneangleforeverytangentvalue,butdoesnotincludepointswherethetangentfunctionisundefined.c○mathcentreJune25,2009 www.mathcentre.ac.uk 8 mc-TY-trig-2009-1