How do you derive the trigonometric formulas for double and half angels for sin, cos, and tan? I.e: How do I derive something like sin(2x)=2(sinx)(cosx)?

1 Answer
Jun 16, 2018

Below

Explanation:

Remember that 2x=x+x

sin(2x)=sin(x+x)=sinxcosx+cosxsinx=2sinxcosx

cos(2x)=cos(x+x)=cosxcosxsinxsinx=cos2xsin2x

Now let 2x=θ so x=θ2

cosθ=cos2(θ2)sin2(θ2)

cosθ=2cos2(θ2)1

2cos2(θ2)=cosθ+1

cos2(θ2)=12(cosθ+1)

cos(θ2)=±12(cosθ+1)

Similarly,

cosθ=12sin2(θ2)

2sin2(θ2)=1cosθ

sin2(θ2)=12(1cosθ)

sin(θ2)=±12(1cosθ)

tan(2x)=tan(x+x)=tanx+tanx1tan2x=2tanx1tan2x

Finally tan(θ2)=sin(θ2)cos(θ2)

tan(θ2)=±12(1cosθ)±12(cosθ+1)

tan(θ2)=1cosθ1+cosθ

tan(θ2)=(1cosθ)(1+cosθ)(1+cosθ)2

tan(θ2)=sinθ1+cosθ