Use De Moivre's Theorem to prove this:

  • Thread starter Thread starter aanandpatel
  • Start date Start date
  • Tags Tags
    Theorem
AI Thread Summary
De Moivre's Theorem is applied to prove that for any n greater than or equal to 1, the equation (1+itanθ)n + (1-itanθ)n = 2cosnθ/cosnθ holds true, provided cosθ ≠ 0. The discussion highlights the initial challenge of converting the expression into modulus-argument form and suggests starting by expressing tanθ as sinθ/cosθ. Participants share insights on simplifying the equation using secθ and complex numbers, which leads to a clearer solution. The exchange emphasizes collaboration and support among users tackling a common HSC question. Overall, the thread illustrates the step-by-step approach to solving the problem using De Moivre's Theorem.
aanandpatel
Messages
16
Reaction score
0
Use De Moivre's Theorem to show that for any n greater that equal to 1

(1+itanθ)n + (1-itanθ)n =2cosnθ/cosnθ

where cosθ ≠ 0


I tried to approach this by converting into modulus argument form but wasn't really sure if that was correct. It's a common New South Wales HSC question but I couldn't find a solution anywhere. Help would be greatly appreciated :)
 
Physics news on Phys.org
The first step would be to convert tanθ into \frac{sin\theta}{cos\theta} and work from there.
 
Thanks a bunch - that helped a lot. Converted it into:
[(secθ)(cosθ+isinθ)]^n + [(secθ)(cosθ-isinθ)]^n and it was easy from there.

Cheers!
 
No problem. Good luck with the HSC, I just finished mine :P
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
View full post »

Similar threads

Z^3-1=0 (express using de moivres theorum)
Replies
10
Views
12K
Complex Roots of Z^n = a + bi: Finding Solutions Using De Moivre's Theorem
Replies
18
Views
2K
Solve de Moivre's Theorem: Sin 3x & Cos 3x
Replies
33
Views
5K
Solving Roots of Unity with De Moivre's Theorem
Replies
18
Views
3K
De Moivre's theorem for Individual sine functions. Need help
Replies
15
Views
3K
Solving z^3 = i using De Moivre's Theorem
Replies
1
Views
3K
De Moivre's Theorem: Find Expansion of cos 5θ
Replies
4
Views
8K
Answer: Proving Sums with De Moivre's Theorem
Replies
4
Views
5K
I Proving De Moivre's Theorem for Negative Numbers?
Replies
8
Views
9K
Factoring the equation z^(n)-1=0 where z is a complex number
Replies
13
Views
31K

Hot Threads

Recent Insights

Back
Top