- #1

- 664

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## 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.