Use it to evaluate
if your question is (3/5+i4/5)[sup]100[/sup]=?the solution is words below
(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)[sup]k[/sup]=cos(kx)+i sin(kx)......(1b)
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)[sup]k[/sup]= (a²+b²)[sup]k[/sup](...) (4)
As for this case,
(3/5)²+(4/5)²=1 angle x could be arcsin(3/5)
the answer is cos[100arcsin(3/5)]+i sin[100arcsin(3/5)]
As far as i can reach
Last edited by George,Y (2006-03-22 00:41:55)
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,
r²Cos²θ = 9/25, r²Sin²θ = 16/25.
Adding the two,
Therefore, Cosθ =3/5, Sinθ =4/5.
[r(Cosθ +iSinθ )]^100=r^100[Cosθ +iSinθ ]^100
Since r=1, r^100=1.
[Cosθ +iSinθ ]^100=Cos100θ +iSin100θ
θ=ArcCos3/5 = ArcSin 4/5
It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.