inverse trigonometric functions：反三角函数

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1、4.7INVERSETRIGONOMETRICFUNCTIONSForaninversetoexistthefunctionMUSTbeone-to-oneAfunctionisone-to-oneifforeveryxthereisexactlyoneyandforeveryythereisexactlyonex.SoIfxand/oryisraisedtoanevenpowerthentheinversedoesnotexistunlessthedomainisrestricted.Theequationy=x2doesnothaveaninversebecausetwodif

2、ferentxvalueswillproducethesamey-value.i.e.x=2andx=-2willproducey=4.Thehorizontallinetestfails.Inordertorestrictthedomain,abasicknowledgeoftheshapeofthegraphiscrucial.Thisisaparabolawith(0,0)asthevertex.Restrictthedomaintotheinterval[0,infinity)tomakeitone-to-one.Nowlet’slookatthetrigfunctions

3、y=sinxy=cosxy=tanxForthegraphofy=sinx,theDomainis(-∞,∞)theRangeis[-1,1]Nota1-1functionSoitcurrentlydoesnothaveaninverseHoweverwecanrestrictthedomainto[-p/2,p/2]Notetherangewillremain[-1,1]Nowit’s1-1!y=sinxTheinverseofsinxorIsdenotedasarcsinxOntheunitcircle:Fortheinversesinefunctionwithangleson

4、lyfrom-p/2top/2ouranswerswillonlybeineitherquadrant1forpositivevaluesandquadrant4fornegativevalues.Findtheexactvalue,ifpossible,y=cosxisnotonetoone,soitsdomainwillalsoneedtoberestricted.y=cosxisnotonetoone,soitsdomainwillalsoneedtoberestricted.Onthisinterval,[0,p]thecosinefunctionisone-to-onea

5、ndwecannowdefinetheinversecosinefunction.y=arccosxory=cos-1xy=arccosxy=cosxOntheunitcircle,inversecosinewillonlyexistinquadrant1ifthevalueispositiveandquadrant2ifthevalueisnegative.Findtheexactvaluefor:y=tanxRememberthattangentisundefinedat-p/2andp/2y=tanxy=arctanxRememberthattangentisundefine

6、dat-p/2andp/2FindtheexactvalueUsingthecalculator.BeinradianmodeArctan(-15.7896)Arcsin(.3456)Arccos(-.6897)Arcsin(1.4535)Arccos(-2.4534)HDub4-7Page349#1-16all,49-67odd