Trigonometric functions - Mathcentre

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1. IntroductionInthisunitweshalluseinformationaboutthetrigonometricratiossine,cosineandtangenttodefine<strong>functions</strong> f(x) = sin x, f(x) = cosxand f(x) = tan x.2. The sine function f(x) = sinxWeshallstartwiththesinefunction, f(x) = sin x.Thisfunctioncanbedefinedforanynumberxusingadiagramlikethis.sin xx1Wetakeacirclewithcentreattheorigin,andwithradius1. Wethendrawalinefromtheorigin,at xdegreesfromthehorizontalaxis,untilitmeetsthecircle,sothatthelinehaslength1.Wethenlookattheverticalaxiscoordinateofthepointwherethelineandthecirclemeet,tofindthevalueof sin x.Theinformationfromthispicturecanalsobeusedtoseehowchanging xaffectsthevalueofsin x. Wecanuseatableofvaluestoplotselectedpointsbetween x = 0 ◦ and x = 360 ◦ ,anddrawasmoothcurvebetweenthem.Wecanthenextendthegraphtotherightandtotheleft,becauseweknowthatthegraphrepeatsitself.x 0 ◦ 45 ◦ 90 ◦ 135 ◦ 180 ◦ 225 ◦ 270 ◦ 315 ◦ 360 ◦sin x 0 0.71 1 0.71 0 −0.71 −1 −0.71 0f(x)1f(x) = sin x−360°0360° 720°x−1c○mathcentreJune25,2009 www.mathcentre.ac.uk 2 mc-TY-trig-2009-1
When x = 0, sin x = 0.Asweincrease xto 90 ◦ , sin xincreasesto1.Asweincrease xfurther,sin xdecreases. Itbecomeszerowhen x = 180 ◦ . Itthencontinuestodecrease,andbecomes−1when xis 270 ◦ .Afterthat sin xincreasesandbecomeszeroagainwhen xreaches 360 ◦ .Wehavenowcomebacktowherewestartedonthecircle,soasweincrease xfurtherthecyclerepeats.Wecanalsousethispicturetoseewhathappenswhen xislessthanzero.Ifwedecrease xfromzero, sin xdecreases.Itbecomes −1when x = −90 ◦ .Thenitbecomeszeroat x = −180 ◦ ,and1at x = −270 ◦ .Itthendecreasesandbecomeszerowhen x = −360 ◦ .Thiscycleisrepeatedifwedecrease xfurther.Fromthispicturewecanseethat,whatevervaluewepickfor x,thevalueof sin xmustalwaysbebetween −1and1. Sothedomainof f(x) = sin xcontainsalltherealnumbers,buttherangeis −1 ≤ sin x ≤ 1. Wecanalsoseethatthefunctionrepeatsitselfevery 360 ◦ . Wecansaythat sin x = sin(x + 360 ◦ ).Wesaythefunctionisperiodic,withperiodicity 360 ◦ .Sometimeswewillwanttoworkinradiansinsteadofdegrees. Ifwehave sin xinradians,itisusuallyverydifferentfrom sin xindegrees. Forexample sin 90 ◦ = 1butinradians sin(90)isabout0.894.Wecanuseatableofvaluesliketheonewehadbeforetoplotagraphof sin xinradians. As 2πradiansisthesameas 360 ◦ thegraphwillbeverysimilartothegraphfor xindegrees,butnowthelabelsontheaxeshavechanged.π π 3π 5π 3πx 0 π4 2 4 4 2sin x 0 0.71 1 0.71 0 −0.71 −1 −0.71 07π42πf(x)1f(x) = sin x−2π02πx4π−13 mc-TY-trig-2009-1 www.mathcentre.ac.uk c○mathcentreJune25,2009
Page 1: TrigonometricfunctionsThesine,cosin
Page 5 and 6: f(x)1f(x) = sin x34−90°0x90°−
Page 7 and 8: Inthiscasewestillhavetwoanglesforso
Page 9: f(x)2f(x) = tan x1−90°0x90°−1

Trigonometric functions - Mathcentre

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