1. ## complex number

If x, y, a and b are real numbers and if x+yi=a/(b+cosA+icosA) , show that (b2-1)(x2+y2) +a2 = 2abx. (i= imaginary number)

I got a feeling that it may have something to do with the formula z+1/z = 2cosx and z-1/z =2isinx. But I just can't figure it out.

2. ## Re: complex number

Typo?

x+yi=a/(b+cosA+icosA)

should be

x+yi=a/(b+cosA+isinA)

3. ## Re: complex number

Originally Posted by Idea
Typo?

x+yi=a/(b+cosA+icosA)

should be

x+yi=a/(b+cosA+isinA)
yeah, you are right, typing mistake^^

4. ## Re: complex number

$\displaystyle b+\cos A+i \sin A = \frac{a}{z}$

$\displaystyle \cos A+i \sin A = \frac{a}{z}-b$

5. ## Re: complex number

Originally Posted by Idea
$\displaystyle b+\cos A+i \sin A = \frac{a}{z}$

$\displaystyle \cos A+i \sin A = \frac{a}{z}-b$
so should I convert cosA to (1/2)(z+1/z) and isinA to (1/2)(z-1/z) ????
Can you explain more?

what is A?

7. ## Re: complex number

I assume A is an arbitrary value (angle) and since it does not appear in the final answer, we should try to eliminate A

using

$\displaystyle \cos ^2A+\sin ^2A=1$

8. ## Re: complex number

Originally Posted by Idea
I assume A is an arbitrary value (angle) and since it does not appear in the final answer, we should try to eliminate A

using

$\displaystyle \cos ^2A+\sin ^2A=1$
I understand that. But I could'nt get cos^2 A +sin^2 A without multiplying (cosA-isinA) to the other side, right?

9. ## Re: complex number

The magnitude of a complex number $\displaystyle w=a+i b$ is defined as

$\displaystyle |w|=\sqrt{a^2+b^2}$