- #1
Moridin
- 692
- 3
Homework Statement
Determine how many roots the equation
[tex](z + \frac{i\sqrt{3}}{2})^{29} = \frac{1+i}{\sqrt{2}}[/tex]
has that are in the first quadrant.
The Attempt at a Solution
I would like to treat the right hand side in the following way.
[tex](z + \frac{i\sqrt{3}}{2})^{29} = \frac{1+i}{\sqrt{2}}[/tex]
[tex]z + \frac{i\sqrt{3}}{2} = (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})^{1/29} = \cos \frac{\pi}{4 \cdot 29} + i \sin \frac{\pi}{4 \cdot 29}[/tex]
It seems reasonable to rewrite the left hand side into
[tex]z + \frac{i\sqrt{3}}{2} = z + i \sin \frac{\pi}{3}[/tex]
Which give us
[tex]z + i \sin \frac{\pi}{3} = \cos \frac{\pi}{4 \cdot 29} + i \sin \frac{\pi}{4 \cdot 29}[/tex]
Now the real part of the LHS must match the real part of the RHS. This means that the real part of z, must be
[tex]Re ~z~ = \cos \frac{\pi}{4 \cdot 29}[/tex]
and that the imaginary part of x must be
[tex]Im ~z ~= \sin \frac{\pi}{4 \cdot 29} - \sin \frac{\pi}{3}[/tex]
From here, I am pretty much lost.