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

Expert Answers
ajju eNotes educator| Certified Educator

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) 



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)