This a question from complex number chapter of fp3
Find the following in the form a+ib, a,b member of real number set:
(a) arccos 4 (b) arcsin 2 (c) arcsin i
I tried to these sums by: x=arccos4, so cosx=4, so e^(ix)-e^-(ix)=8
The answers:
(a) My answer:+/- i arcosh 4
Book's answer: 2m*pi +/- i arcosh 4
(b) My answer: +/- arcosh 2(i am not sure how i get that)
Book's answer: (4m+1)*pi/2 +/- i arcosh 2
(c) My answer: i ln ((-2+/- sqrt 5)/2))
Book's answer: n*pi + i arsinh [(-1)^n]
So I gathered that I am missing something major here. Please anyone tell me the correct method to do these sums, thus enabling me to try it out myself. Just tell me the method, please This is my first sums of this kind:woo:
P.s. i=imaginary.
My ideas were coming randomly, hence the array of posts
I am not actually familiar with multi-valuation of imaginary exponential funtion, but I think i can solve the equation in the trigonometric form. But I am not sure how sin x=0 can lead to (4m+1)*pi/2 as the value of x, especially when 3/2 pi is a solution of x. Can you give me a site which addresses how to get the general formula for cos x=0, sinx =0 , sin x=c etc. I will be perfectly happy then
The key to understanding these is understanding the multi-valued logarithm function. Until you understand that, you're going to have problems throughout Complex Analysis. Now
That formula is a big deal because both logarithm and the root object are multivalued:
Then:
This then bifurcates into two infinite sets of values:
Where in this case and