Trigonometric Identities - Miami

1 Trigonometric IdentitiesSum and Difference Formulassin (x+y) = sinxcosy+ cosxsinysin (x y) = sinxcosy cosxsinycos (x+y) = cosxcosy sinxsinycos (x y) = cosxcosy+ sinxsinytan (x+y) =tanx+tany1 tanxtanytan (x y) =tanx tany1+tanxtanyHalf-Angle Formulassin 2= 1 cos 2cos 2= 1+cos 2tan 2= 1 cos 1+cos tan 2=1 cosxsinxtan 2=sin 1+cos Double-Angle Formulassin 2 = 2 sin cos cos 2 = cos2 sin2 tan 2 =2 tan 1 tan2 cos 2 = 2 cos2 1cos 2 = 1 2 sin2 Product-to-Sum Formulassinxsiny=12[cos (x y) cos (x+y)]cosxcosy=12[cos (x y) + cos (x+y)]sinxcosy=12[sin (x+y) + sin (x y)]Sum-to-Product Formulassinx+ siny= 2 sin(x+y2)cos(x y2)sinx siny= 2 sin(x y2)cos(x+y2)cosx+ cosy= 2 cos(x+y2)cos(x y2)cosx cosy= 2 sin(x+y2)sin(x y2)The Law of SinessinAa=sinBb=sinCcSuppose you are given two sides,a,band the angleAopposite the sideA.

2 Theheight of the triangle ish=bsinA. Then1. Ifa < h, thenais too short to form a triangle, so there is no Ifa=h, then there is one Ifa > handa < b, then there are two distinct Ifa b, then there is one Law of Cosinesa2=b2+c2 2bccosAb2=a2+c2 2accosBc2=a2+b2 2abcosC