Trigonometric functions - Mathcentre

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Tocomparethetwographs,wecankeepthesamescaleonthe x-axisandplotbothgraphs.f(x)1x in degrees060xx in radiansWith xindegrees,thefunction f(x) = sin xhasnotreached1bytheright-handsideofthegraph,butwith xinradiansthefunctionhasoscillatedseveraltimes.Sothesearequitedifferent<strong>functions</strong>.Sometimes,insteadoffindingthesineofanangle,wewanttoworkbackwards. Wewanttofindananglewhosesineis,say, 3 4 . Sowewanttodefineanewfunctiontogiveustheinversesineofthenumber. Wewanttofinda<strong>functions</strong>uchthat f −1 (x) = ywhenever f(y) = x. Inourcase,wewant sin x = 3 4 ,sothatweshallwanttohave sin−1 ( 3 4 ) = x.Nowthismightseemtobeaproblematfirstbecause,ifwelookbackatourgraph,weseethattherearelotsofangleswith sin x = 3 4 .f(x)1f(x) = sin x34−360°0360° 720°x−1Wecannotdefineafunctiontotelluswhattheinversesineof 3 4 shouldbeifthereisachoiceofvaluesfor f −1 (x). Togetaroundthisproblem,weneedtorestrictthedomainofourfunctionf(x) = sin xsothatwehaveonlyapartofthegraphthatgivesusoneangleforeachsinevalue.Thishappensifwecutourdomaindownto −90 ◦ ≤ x ≤ 90 ◦ ,or −π ≤ x ≤ πifweworkinradians.c○mathcentreJune25,2009 www.mathcentre.ac.uk 4 mc-TY-trig-2009-1
f(x)1f(x) = sin x34−90°0x90°−1Wesaythatourfunction f(x) = sin xhasdomain −90 ◦ ≤ x ≤ 90 ◦ andthatithasaninverse,f −1 (x) = sin −1 x. Thisinversefunctionisalsowrittenas arcsin x.So,iftheangle xliesintherange −90 ◦ ≤ x ≤ 90 ◦ and sin x = 3 4 ,wesay x = sin−1 ( 3 4 ).Youcanuseyourcalculatortoworkoutinversesines.Key PointThefunction f(x) = sin xhasallrealnumbersinitsdomain,butitsrangeis −1 ≤ sin x ≤ 1.Thevaluesofthesinefunctionaredifferent,dependingonwhethertheangleisindegreesorradians.Thefunctionisperiodicwithperiodicity360degreesor2πradians.Wecandefineaninversefunction,denoted f(x) = sin −1 xor f(x) = arcsin x,byrestrictingthedomainofthesinefunction.3. The cosine function f(x) = cos xWeshallnowlookatthecosinefunction, f(x) = cosx. Thisfunctioncanbedefinedforanynumber xusingadiagramlikethis.xcos x 15 mc-TY-trig-2009-1 www.mathcentre.ac.uk c○mathcentreJune25,2009
Page 1 and 2: TrigonometricfunctionsThesine,cosin
Page 3: When x = 0, sin x = 0.Asweincrease
Page 7 and 8: Inthiscasewestillhavetwoanglesforso
Page 9: f(x)2f(x) = tan x1−90°0x90°−1

Trigonometric functions - Mathcentre

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