Derive a formula for cos3theta, in terms of costheta and sintheta or just costheta. ?????

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ajju | College Teacher | (Level 1) Adjunct Educator

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we can split 3θ as (2θ + θ)

then cos(3θ)=cos(2θ+θ) 

and the addition rule of cosine: cos(a+b)=cos(a)cos(b)-sin(a)sin(b) 

Therefore: 

cos(3θ)=cos(2θ)cos(θ)-sin(2θ)sin(θ) 

Put these values of sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) =cos^2(θ)-sin^2(θ):

then we get: 

cos(3θ)=[cos^2(θ)-sin^2(θ)]cos(θ) - (2sin(θ)cos(θ)) sin(θ)


cos(3θ)=cos^3(θ)-sin^2(θ)cos(θ) - 2sin^2(θ)cos(θ) 


cos(3θ)=cos^3(θ) - 3sin^2(θ)cos(θ)

cos(3θ)= cos(θ)[cos^2(θ) - 3sin^2(θ)]

put the value of sin^2(θ) = 1 - cos^2(θ)

then:  cos(3θ)= cos(θ)[cos^2(θ) - 3(1 - cos^2(θ))]

cos(3θ)= cos(θ)[cos^2(θ) - 3 + 3cos^2(θ)]

cos(3θ)= cos(θ)[4cos^2(θ) - 3]

cos(3θ)= 4cos^3(θ) - 3cos(θ)

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