So I have an illustrative picture of all of the roots of one complex number (the exponent is random). I have to find this number $z$, if we know that these roots are on a circle, that's diameter is $3.4$ and degree between the real axis and the positive direction is $-61$ degrees.
Here is the illustrative picture that I have:
I got the answer that the complex number is $z=0.824+1.487i$. but all I really need is $a$
Are my calculations right?
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Assuming I understand the question correctly, we have: $$w=z^{\frac{1}{6}}=\frac{3.4}{2} \cdot (\cos{(-61)}+i \cdot \sin{(-61)}) = 0.82418-1.48685i$$ Which is what you have. But this is $w=z^{\frac{1}{6}}$ ; $$\therefore z = w^6 = 24.00534-2.52306i$$
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I think you're close.
If I take $\sqrt{0.824^2+1.487^2}$, this equals $1.7$, so that checks.
However, if your angle is $-61^{\circ}$, I'd expect the imaginary part to be negative (the angle lies in the fourth quadrant).
You have $\tan ^{-1} 1.487/0.824 = 61^{\circ}$, so the magnitudes are right.
Do you see how to fix it?
That's one of the roots. Now for the other five roots, you add $60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}$ to the angle.
Change the $+$ to a $-$ and you're golden.
Presumably you have calculated $$x = 1.7\cos(-61^\circ), y = 1.7\sin(-61^\circ),$$ which does indeed give the answer you suggest.