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I don't understand how this can form can work: $r(\cos(x)-i\sin(x))$. I saw it in my textbook. Surely there should be a "$+$" rather than a "$-$" between the $\cos$ and the $\sin$. If the imaginary part was negative then surely that would be accounted for by the argument which in this case I have written as "$x$".

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    $\begingroup$ you are right, it should be $+$ $\endgroup$
    – Idonknow
    Sep 14 '14 at 15:07
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    $\begingroup$ Perhaps they were writing $\bar z$, the complex conjugate of $z$? $\endgroup$ Sep 14 '14 at 15:09
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    $\begingroup$ You can write a complex number either as $r(\cos x - i \sin x)$ or as $r(\cos\theta + i \sin\theta)$, simply by substituting $x=-\theta$. $\endgroup$
    – Jack Lee
    Sep 14 '14 at 15:09
  • $\begingroup$ Where in what book? $\endgroup$ Sep 14 '14 at 15:22
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    $\begingroup$ IF you gave an exact quote from the textbook, including context, we might be able to help. $\endgroup$ Sep 14 '14 at 15:23
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Perhaps you could use the fact that $-\sin (x) = \sin (-x)$ and $\cos (x) = \cos (-x)$, then your expression becomes $r(\cos (-x) + i \sin (-x))$.

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