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Quantum Optics for Engineers by F.J. Duarte

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Appendix F: Trigonometric Identities

F.1  Trigonometric Identities

Here, we list a number of useful trigonometric identities applied in various places throughout this book:

sin2ϕ+cos2ϕ=1

(F.1)

 sin (ϕ)=sinϕ

(F.2)

 cos (ϕ)= cos ϕ

(F.3)

 sin (ϕ+ψ)= sinϕcosψ+ cosϕsinψ

(F.4)

 sin (ϕψ)= sinϕcosψcosϕsinψ

(F.5)

cos (ϕ+ψ)= cosϕcosψsinϕsinψ

(F.6)

cos (ϕψ)= cosϕcosψ+sinϕsinψ

(F.7)

 sin 2ϕ=2sinϕcosϕ

(F.8)

cos2ϕ=cos2ϕsin2ϕ

(F.9)

cos 2ϕ=1sin2ϕ

(F.10)

cos 2ϕ=2cos2ϕ1

(F.11)

2sin2ϕ=1cos2ϕ

(F.12)

2cos2ϕ=1+cos2ϕ

(F.13)

cosϕ+cosψ=2 cos (ϕψ)cos(ϕ+ψ)

(F.14)

cosϕcosψ=2sin(ϕψ)sin(ϕ+ψ)

(F.15)

cosϕ+cosψ=2sin(ϕψ)sin(ϕ+ψ)

(F.16)

cosϕ

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