Class notes math trignomatrey

Fundamental Identities Triple Angle Identities Product into Sum or Difference
i sin2 θ + cos 2 θ = 1 sin 3θ = 3 sinθ − 4 sin3 θ 2 sin α cos β = sin α+ β + sin α− β
ii 1 + tan2 θ = sec 2 θ cos 3θ = 4 cos3 θ − 3 cos θ 2 cos α sin β = sin α+ β − sin α− β
2 2 3
iii 1 + cot θ = cosec θ 3tanθ − tan θ
tan 3θ = 2 cos α cos β = cos α+ β + cos α− β
1 − 3 tan2 θ
− 2 sin α sin β = cos α+ β − cos α− β
cos α+ β = cos α cos β − sin α sin β
cos α− β = cos α cos β + sin α sin β Double Angle Identities
sin α+ β = sin α cos β + cos α sin β sin 2 α = 2 sin α cos α
sin α− β = sin α cos β − cos α sin β
cos 2 α = cos 2 α − sin2 α
tan α + tan β
tan α+ β = cos 2 α = 2 cos2 α − 1 = 1 − 2 sin2 α
1 − tan α tan β
2 tan α
tan α − tan β tan 2 α =
tan α− β = 1 − tan2 α
1 + tan α tan β

Half Angle Identities
Sum or Difference into Product
θ
P+Q P−Q 1 + Cos θ = 2 cos 2
sin P + sin Q = 2 sin cos 2
2 2
θ
P+Q P−Q 1 − Cos θ = 2 sin2
sin P − sin Q = 2 cos sin 2
2 2
P+Q P−Q θ 1 − cos θ
Cos P + Cos Q = 2 cos cos tan = ±
2 2 2 1 + cosθ
P+Q P−Q
Cos P − Cos Q = − 2 sin sin
2 2