Derive formulas for the following in terms of functions of 2theta and then of theta. 1. sin4theta 2. cos4theta 3. tan4theta
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Given : Sin 4θ , Cos4θ , Tan 4θ
To Find : formulas for the following in terms of functions of 2θ
Solution:
Sin 4θ = Sin(2θ + 2θ)
Sin (A + B) = SinACosB + CosASinB
= Sin2θCos2θ + Cos2θSin2θ
= 2Sin2θCos2θ
Sin 4θ = 2Sin2θCos2θ
Cos4θ =Cos(2θ + 2θ)
Cos (A + B) = CosACosB -SinASinB
= Cos2θCos2θ -Sin2θSin2θ
= Cos²2θ - Sin²2θ
Cos4θ = Cos²2θ - Sin²2θ
Tan(A + B) = (Tan A + TanB)/(1 - TanA TanB)
Tan 4θ = Tan (2θ + 2θ)
Hence Tan 4θ = 2 Tan2θ / (1 - Tan²2θ)
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