DeMoivre's theorem: cos3θ in powers of cosθ

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In summary, use of de Moivre's Theorem to express cos3θ in powers of cosθ involves expanding the brackets using the binomial theorem and eliminating the imaginary part by using i^2=-1. The final answer is (cosθ)^3 - 3cosθ(sinθ)^2.
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ZedCar
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Homework Statement



Use de Moivre's Theorem to express cos3θ in powers of cosθ

Homework Equations



z^n = [r(cosθ + isinθ)]^n = r^n (cos(nθ) + i sin(nθ))

The Attempt at a Solution



cos3θ = Re(cos3θ +isin3θ) = Re[(cosθ +isinθ)^3]

I've then expanded the brackets using binomial theorem and got;

(cosθ)^3 + 3[(cosθ)^2][isinθ] + 3(cosθ)[(isinθ)^2] + (isinθ)^3

So (cosθ)^3 is the real part and 3[(cosθ)^2][isinθ] + 3(cosθ)[(isinθ)^2] + (isinθ)^3 the imaginary part.

If anyone has any suggestions...

Thank you
 
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  • #2
use i^2=-1 to write (cosθ)^3 + 3[(cosθ)^2][isinθ] + 3(cosθ)[(isinθ)^2] + (isinθ)^3 in a+bi form
 
  • #3
Okay, so doing that I'm getting:

(cosθ)^3 - 3cosθ(sinθ)^2 + 3i(cosθ)^2 (sinθ) - i(sinθ)^3

So the real part is (cosθ)^3 - 3cosθ(sinθ)^2

the imaginary part 3i(cosθ)^2 (sinθ) - i(sinθ)^3
 
  • #4
Yes, though the imaginary part is usually defined to not have the i
also the question asked for an answer in cosθ so eliminate sinθ
 
  • #5
Okay, thanks. I've got the answer.
 

FAQ: DeMoivre's theorem: cos3θ in powers of cosθ

What is DeMoivre's theorem?

DeMoivre's theorem is a mathematical theorem that relates powers of complex numbers to their trigonometric functions. It states that for any complex number z and any positive integer n, (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ).

How is DeMoivre's theorem used to find cos3θ in powers of cosθ?

To find cos3θ in powers of cosθ, we can use DeMoivre's theorem to expand (cos θ + i sin θ)^3. This results in cos3θ + i sin3θ = cos^3θ + 3i cos^2θ sinθ - 3cosθ sin^2θ - i sin^3θ. By equating the real and imaginary parts, we can express cos3θ in terms of cosθ and sinθ.

What is the significance of cos3θ in powers of cosθ?

Finding cos3θ in powers of cosθ allows us to express a higher degree trigonometric function (cos3θ) in terms of a lower degree trigonometric function (cosθ). This can be useful in simplifying complex trigonometric expressions and solving equations involving trigonometric functions.

Can DeMoivre's theorem be applied to other trigonometric functions?

Yes, DeMoivre's theorem can also be applied to find powers of other trigonometric functions, such as sin θ and tan θ. The only difference is that the resulting expression will involve both real and imaginary parts.

How does DeMoivre's theorem relate to the polar form of a complex number?

The polar form of a complex number z is expressed as z = r(cos θ + i sin θ), where r is the modulus of z and θ is the argument of z. DeMoivre's theorem can be used to raise this polar form to any positive integer power, resulting in z^n = r^n(cos nθ + i sin nθ). This is a useful tool in computing powers of complex numbers.

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