Answer to Question #91337 in Algebra for Ra

Question #91337
Use DeMoivre's theorem to show that cos3θ=4cos³θ-3cosθ
1
Expert's answer
2019-07-02T12:04:03-0400

De Moivre's theorem states that for any real number θ and integer n it holds that


(cos(θ)+isin(θ))n=cos(nθ)+isin(nθ).\big(\cos(\theta) + i \sin(\theta)\big)^n = \cos(n\theta) + i\sin(n\theta).

Let n = 3, then


(cos(θ)+isin(θ))3=cos(3θ)+isin(3θ).\big(\cos(\theta) + i \sin(\theta)\big)^3 = \cos(3\theta) + i\sin(3\theta).


Then use the Binomial theorem to expand the terms in the brackets:

cos3(θ)+3isin(θ)cos2(θ)3sin2(θ)cos(θ)isin3(θ)=cos(3θ)+isin(3θ).\cos ^3(\theta )+3 i \sin (\theta ) \cos ^2(\theta )-3 \sin ^2(\theta ) \cos (\theta )-i \sin ^3(\theta ) = \cos(3\theta) + i\sin(3\theta).


Thus, equating the real and imaginary parts, we get

cos3(θ)3sin2(θ)cos(θ)=cos(3θ),\cos ^3(\theta )-3 \sin ^2(\theta ) \cos (\theta ) = \cos(3\theta),3sin(θ)cos2(θ)sin3(θ)=sin(3θ).3 \sin (\theta ) \cos ^2(\theta )-\sin ^3(\theta ) = \sin(3\theta).

Then

cos(3θ)=cos3(θ)3(1cos2(θ))cos(θ)=4cos3(θ)3cos(θ).\cos(3\theta) = \cos ^3(\theta )-3 \big(1-\cos^2(\theta )\big) \cos (\theta ) = 4\cos ^3(\theta )-3\cos (\theta ).

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment