Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2006-03-22 22:12:40

coolwind
Member

Offline

### How to use DeMoivre's formula?

Use it to evaluate
(3/5+j4/5)^100 ?

Thaks~~

## #2 2006-03-22 23:40:34

George,Y
Super Member

Offline

### Re: How to use DeMoivre's formula?

if your question is (3/5+i4/5)100=?the solution is words below

Key theories:

(1) (cosa+i sina)(cosb+i sinb)
= cosa cosb - sina sinb + (sina cosb + cosa sinb)i

= cos(a+b)+ sin(a+b)i

(cosa+i sina)(cosa+i sina)(cosa+i sina) = (cos2a+isin2a)(cosa+i sina) =cos3a+isin3a

Similarly, (cosx+i sinx)k=cos(kx)+i sin(kx)......(1b)

cos(x)²+sin(x)²=1    (3)

any a+bi can be expressed as
a+bi = (a²+b²)(a+bi)/(a²+b²) = (a²+b²) (a/(a²+b²)+b/(a²+b²) i)
a/(a²+b²) and b/(a²+b²) satisfy cos(x) and sin(x) in equation(3) . Actually we need  not point out angle x explicitly sometimes, besides, i forgot the formula!

(a+bi)k= (a²+b²)k(...)  (4)

As for this case,

(3/5)²+(4/5)²=1 angle x could be arcsin(3/5)

As far as i can reach

Last edited by George,Y (2006-03-22 23:41:55)

X'(y-Xβ)=0

## #3 2006-03-23 02:17:43

ganesh
Moderator

Offline

### Re: How to use DeMoivre's formula?

Yes, I guess the question is (3/5 + 4/5i).
This is of the form a+bi. We got to convert this to polar form. That is r(Cosθ +iSinθ).
Equating the real and imaginary parts,
rCosθ=3/5, rSinθ=4/5.
Squaring both,
r²Cos²θ  = 9/25, r²Sin²θ = 16/25.