4 sin theta into sin pi by 3 + theta into sin 2 pi by 3 + theta equals to sin 3
theta
Answers
Answer:
4Sinθ * Sin(π/3 + θ) * Sin(2π/3 + θ) = Sin3θ
Step-by-step explanation:
4 sin theta into sin pi by 3 + theta into sin 2 pi by 3 + theta equals to sin 3
theta
4Sinθ * Sin(π/3 + θ) * Sin(2π/3 + θ) = Sin3θ
LHS = 4Sinθ * Sin(π/3 + θ) * Sin(2π/3 + θ)
Sinx = Sin(π - x) => Sin(2π/3 + θ) = Sin(π/3 - θ)
= 4Sinθ * Sin(π/3 + θ) * Sin(π/3 - θ)
Sin(A + B)* Sin(A - B) = Sin²A - Sin²B
= 4Sinθ * (Sin²π/3 - Sin²θ)
Sin²π/3 = 3/4
= 4Sinθ * (3/4 - Sin²θ)
= 3Sinθ - 4 Sin³θ
RHS = Sin3θ
= Sin(2θ + θ)
Sin(A+B) = SinA CosB + CosA SinB
= Sin2θCosθ + cos2θSinθ
Using Sin2θ = 2SinθCosθ & cos2θ = cos²θ - Sin²θ
= 2SinθCosθCosθ + (cos²θ - Sin²θ)Sinθ
= Sinθ ( 2cos²θ + cos²θ - Sin²θ)
= Sinθ(3Cos²θ - Sin²θ)
Cos²θ = 1 - Sin²θ
= Sinθ(3 - 3Sin²θ - Sin²θ)
= Sinθ(3 - 4Sin²θ)
= 3Sinθ - 4 Sin³θ
LHS = RHS
4Sinθ * Sin(π/3 + θ) * Sin(2π/3 + θ) = Sin3θ