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I want to find the Argument of $z = -\sqrt{2 - \sqrt{3}} + i\sqrt{2 + \sqrt{3}}$ where $z$ is a complex number of the form $z = a + bi$.

I find that the modulus is $2$, but am having trouble simplifying $\theta = \arctan\left(\frac{\sqrt{2 + \sqrt{3}}}{-\sqrt{2 - \sqrt{3}}}\right)$. I can put the squareroot sign over the whole fraction, but that still doesn't really help me get an actual number for theta.

ALSO: It is a rule that I add $\pi$ to theta if $a < 0$. Why? Looking at the diagram of the triangle form by the complex vector $z$ and the real axis, the 'triangle' is in the second quadrant. Hence, the theta we find with $\theta = \tan^{-1}(\frac{b}{a})$ is the angle closes to the real axis on the left-side. However, we measure angle going counter-clockwise. Shouldn't we do the following computation to get the angle going clock-wise: $\pi - \tan^{-1}(\frac{b}{a})$?

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Note

$$\frac{1}{2}\sqrt{2 - \sqrt{3}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{1 - \cos \frac{\pi}{6}}{2}} = \sin \frac{\pi}{12} $$

and similarly $$\frac{1}{2}\sqrt{2 + \sqrt{3}} = \cos \frac{\pi}{12}.$$ Therefore $$z = 2\left(-\sin \frac{\pi}{12} + i\cos \frac{\pi}{12}\right) = 2e^{i(\frac{\pi}{2} + \frac{\pi}{12})} = 2e^{i\frac{7\pi}{12}}.$$

Since $-\pi < \frac{7\pi}{12} \le \pi$, the principal argument of $z$ is $\frac{7\pi}{12}$.

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Note that: $$ \dfrac{\sqrt{2+\sqrt{3}}}{-\sqrt{2-\sqrt{3}}}=-\sqrt{\dfrac{2+\sqrt{3}}{2-\sqrt{3}}}=-\sqrt{\dfrac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}}=-(2+\sqrt{3}) $$ since $2+\sqrt{3}=\tan \left(\dfrac{5 \pi}{12}\right)$, and $\theta$ must be in the second quadrant we have $\theta =\left(\dfrac{7 \pi}{12}\right)$

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  • $\begingroup$ Not true. The arctangent function only returns values from $-\pi/2$ to $\pi/2$ so what you really mean is $\theta = -5\pi/12 + n\pi$ $\endgroup$ – Dylan Feb 2 '15 at 21:12
  • $\begingroup$ Thanks! I edit and correct my mistake $\endgroup$ – Emilio Novati Feb 2 '15 at 21:19
  • $\begingroup$ @EmilioNovati How did you know that $2 + \sqrt{3} = \tan(\frac{5\pi}{12})$? If we slice the unit circle in $12$ segments, each interior angle at the origin is $30deg$. $\endgroup$ – Hlepkit Feb 3 '15 at 19:56
  • $\begingroup$ So, if you slice half a circle ( that is$\pi$) in $12$ segments, each interior angle is $15$ deg. $\endgroup$ – Emilio Novati Feb 3 '15 at 20:00
  • $\begingroup$ Mmm..okay, which means that $\frac{5\pi}{12} = 75deg$, but i'm still not seeing the $2 + \sqrt{3}$ part. Also, I know I can use trig identities here, but I know there is a way to do it without them. $\endgroup$ – Hlepkit Feb 3 '15 at 20:06
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These numbers may look pretty complicated but notice that if we square both sides a lot can be simplified.

$$ z^2 = \big((2 - \sqrt{3}) - (2 + \sqrt{3})\big) - 2\sqrt{(2 - \sqrt{3})(2 + \sqrt{3})}\,i = -2\sqrt{3} - 2i = 4 \left(\cos \left(\frac{7\pi}{6}\right) + i\sin \left(\frac{7\pi}{6}\right) \right) $$

So now we know the argument of $z^2$ is $\frac{7\pi}{6}$, we can relate this to the argument of $z$ according to de Moivre's theorem

$$ (\cos \theta + i\sin \theta)^2 = \cos (2\theta + 2n\pi) + i\sin (2\theta + 2n\pi) $$

If $\theta$ is the argument of $z$, the argument of $z^2$ is $$2\theta + 2n\pi = \frac{7\pi}{6}$$

Solving that, we get

$$\theta = \frac{7\pi}{12} + n\pi = \frac{7\pi}{12}, \frac{19\pi}{12} $$

We got two answers, because one is the argument of $z$, and the other is the argument of $-z$.

Since $\cos \theta < 0$ and $\sin \theta > 0$, $\theta$ must lie on the second quadrant (between $\pi/2$ and $\pi$), therefore $ \theta = \frac{7\pi}{12}$

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