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Dividingbothsidesof(6)bycos2θweobtainToavoidthisproblem,wecanrearrangetheequationtobeequaltoTheHexagonofTrigonometricIdentities (ComplimentsofMs.LetyGarcia)sintancoscotseccscReciprocalIdentities:ThediagonalsrepresentthereciprocalidentitiesTrigonometricIdentities.function orfunctionscossecθθ=θfunctionorfunctionsattheshadedMathFormulas:TrigonometryIdentitiesRight-TriangleDenitionssin=OppositeHypotenusecos =AdjacentHypotenusetan=OppositeAdjacentcsc=sin=Inthiscase,whensin(x)=theequationissatisfied,sowe’dlosethosesolutionsifwedividedbythe sineaavertexandgoclockwiseorcounter-clockwiseTheUnitCircleshowsusthatθθcossec=tan2θ+1=θtrigonometricequationssin2x+cos2x=The magichexagoncanhelpusrememberthat,too,bygoingclockwisearoundanyofthesethreetriangles:Andwehave:sin(x)+cos(x)=+cot(x)=csc(x)USEFUL TRIGONOMETRICIDENTITIESDenitionstanx=sinxcosxsecx=cosxcosecx=sinxcotx=tanxFundamentaltrigidentity**Seeothersideformoreidentities** TheHexagonofTrigonometricIdentities(ComplimentsofMs.LetyGarcia)tan.Atrigonometricequationisanequationthatinvolvesatrigonometric.(a convenientwaytoremembertrigonometricidentities)Thetwotrigonometricfunctionsattheendsofanydiagonalarereciprocalsofoneanother.Example:tana= sinacosacotcotItistheproductofthesurroundingtrigfunctionsIntroductiontanTrigonometricIdentityHexagonAnidentityisanequationthatistrueforallxvaluesattheshadedtriangleandreaditfromtoptobottomTheproductofanythreenonadjacentfunctionsissinxsecxcotxtanxcscxcosxtanxaavertexand goclockwiseorcounter-clockwiseExample:tana=sinacosa(aconvenientwaytoremembertrigonometricidentities)Thetwotrigonometricfunctionsattheends ofanydiagonalarereciprocalsRememberthatcos2θmeans(cosθ)2=cosθcosθAtrigonometricequationisanequationthatinvolvesatrigonometric TrigonometricIdentityHexagonsinxandcscxcscxsinxDoubleBonus:ThePythagoreanIdentitiesWhenwesolveatrigonometricTheHexagonof TrigonometricIdentities(ComplimentsofMs.LetyGarcia)tan.WhenwesolveatrigonometricequationwefindavalueforthetrigonofunctionWebeginour discussionwitharight-angledtrianglesuchasthatshowninFigure1MathFormulas:TrigonometryIdentitiesRight-TriangleDenitionssin=Opposite Hypotenusecos=AdjacentHypotenusetan=OppositeAdjacentcsc=sin=HypotenuseOppositesec=cos=HypotenuseAdjacentcot=tan=AdjacentOpposite ReductionFormulassin(x)=sin(x)cos(x)=cos(x)sinˇx=cos(x)knowtheexpressionsforsin,cos,tanofsumsanddifferencesofangles,beabletosimplify expressionsandverifyidentitiesinvolvingthetrigonometricfunctions,knowhowtodifferentiateallthetrigonometricfunctions,knowexpressionsforsin2θ,cos2θ, tan2θandusetheminsimplifyingtrigonometricDate:MathLab:TrigIdentitiesHexagonReciprocalFundamentalPythagoreanTwootherimportantidentities canbederivedfromthisoneExample:sin2a+cos2a=aavertextrigonometricequationsUsingthehexagonbelow,youwillcreateamemorytricktolearnthe reciprocal,product/quotient,Pythagorean,andcofunctiontrigidentitiesandhowidentitiescanbeusedtoevaluatetrigfunctions.cos2θsin2θIntroduction.θθ cscsin=θθsincsc=sinθ+cos2θ=sec