Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

CHEMMIB

  • 2 years ago

Use De Moivre's theorem to express cos 5θ and sin 5θ in terms of sin θ and cos θ.

  • This Question is Closed
  1. LolWolf
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    So, by de Moivre's we know: \[ (\cos \theta+i \sin\theta)^n=\cos(n\theta)+i\sin(n\theta) \]Therefore, we know: \[ \Re\left(\left(\cos\theta+i\sin\theta\right)^5\right)=\cos(5\theta)\\ \Im\left(\left(\cos\theta+i\sin\theta\right)^5\right)=\sin(5\theta) \]Expand the previous identities, and find the real part to give a closed-form expression for \(\cos\theta\), it follows similarly (for the imaginary part) for \(\sin\theta\).

  2. CHEMMIB
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Would my answer then be (cos θ + i sin θ)^5 = cos 5θ + i sin 5θ?

  3. LolWolf
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Although the original answer I gave *is* a correct answer. It should not be the final statement, it was more of a hint. Let's solve for \(\cos(\theta)\) and leave the other for you. Using binomial theorem: \[ \left(\cos\theta+i\sin(\theta)\right)^5=\\ \cos^5\theta+C_1i(\sin\theta)(\cos^4\theta)+C_2(i^2\sin^2\theta)(\cos^3\theta)+\cdots+C_5(i^5\sin^5\theta) \]We can ignore all of the odd exponents (excluding the first), since we are looking for the real part, and, thus, we get: \[ \cos^5\theta-10(\sin^2\theta)(\cos^3\theta)+5(\sin^4\theta)(\cos\theta)=\cos(5\theta) \]

  4. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.