Solving Complex Number Homework: No Value for x

[zorro]

Homework Statement

If cosx - isin2x and sinx - icos2x are conjugates of each other, then what is the value of x?

The Attempt at a Solution

Since the given complex numbers are conjugates of each other,
their modulus must be same.
i.e.

cos²x + sin²2x = sin²x + cos²2x
cos2x = cos4x
On solving, I got x=nπ/3 or x = nπ

But the answer given is- No value of x.

Please help

 

[hunt_mat]

if $$z_{1}=\cos x - i\sin 2x,\quad z_{2}=\sin x - i\cos 2x$$ Then, $$\bar{z}_{1}=z_{2}$$ and we obtain:
$$\cos x - i\sin 2x=\sin x +i\cos 2x$$
Equate and and imaginary parts...

 

[zorro]

Thanks for your reply.
I already got the solution that way.
I would like to find out what is the problem with my solution.

 

[willem2]

if the two numbers are conjugates, there modulus must be the same is true.
If two complex numbers have the same modulus, they are conjugates is not true.

 

[zorro]

I completely agree with you that its converse is not true.
But its already given that the numbers are conjugates of each other so their moduli must be same. ( I'm not applying any converse here)

 

[Mentallic]

It's not enough for their moduli to be the same, their angles must be negatives of each other too. And since it gives you that they're conjugates, then you just need to equate their real parts, and the negative of their imaginary parts.

 

[zorro]

Thanks!