Trigonometric Identities 1

TRIGONOMETRICIDENTITIESApplyingPythagoras’theoremtoacircleofunitradiusyieldsthefollowingidentity:(1)Cos2(x)+Sin2(x)=1.Usingtherotationmatrixtorotatebyxdegreesfollowedbyydegreesyieldsthefollowingidentities:(2)Cos(x+y)=Cos(x)Cos(y)–Sin(x)Sin(y)(3)Cos(x-y)=Cos(x)Cos(y)+Sin(x)Sin(y)(4)Sin(x+y)=Sin(x)Cos(y)+Cos(x)Sin(y)(5)Sin(x-y)=Sin(x)Cos(y)–Cos(x)Sin(y).Substitutingx=yintoequations(2)and(4)yieldthefollowing:(6)Cos(2x)=Cos2(x)–Sin2(x)(7)Sin(2x)=2Sin(x)Cos(x)Lookat(1)and(6)rewrittenwiththeCos(2x)ontheR.H.S.(1)Cos2(x)+Sin2(x)=1.(6)Cos2(x)–Sin2(x)=Cos(2x)(1)+(2)yieldstheidentity:(8)2Cos2(x)=1+Cos(2x),whichcanbewrittenas:(9)Cos2(x)=2))2(1(xCosFocusingonthesamepairofequations:(1)Cos2(x)+Sin2(x)=1.(6)Cos2(x)–Sin2(x)=Cos(2x)Subtracting(1)–(6)yields:(10)2Sin2(x)=1–Cos(2x),whichmaybewrittenas:(11)Sin2(x)=2))2(1(xCosTRIGONOMETRICIDENTITIESTangentformulae:Equations(4)and(2)canbeusedtoderiveformulaeforTangentanglesums:Tan(x+y)=Sin(x+y)Cos(x+y)=Sin(x)Cos(y)+Cos(x)Sin(y)Dividingtop&bottombyCos(x)Cos(y)yields:Cos(x)Cos(y)-Sin(x)Sin(y)(12)Tan(x+y)=Tan(x)+Tan(y)1-Tan(x)Tan(y)Replacingywith–y,notingthatTan(-y)=-Tan(y)=yields:(13)Tan(x-y)=Tan(x)-Tan(y)1+Tan(x)Tan(y)