anonymous
  • anonymous
please help
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
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anonymous
  • anonymous
First of all, you know the Argand Diagram is drawn like so.|dw:1441202572576:dw| Let's do the first part first. The point \(1 + i\sqrt{3}\) has x = 1, y = \(\sqrt{3}\), right?
anonymous
  • anonymous
yes

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anonymous
  • anonymous
To figure out what \(|z - 1 - i\sqrt3| \le 1\) is, let \(z = x + yi\). Substituting in, \(|z - 1 - i\sqrt3| \le 1 \implies |(x - 1) + (y - \sqrt{3})i| \le 1\). Do you think you can simplify this inequality into a form that you are familiar with?
anonymous
  • anonymous
its a circle with centre \[1+i \sqrt{3}\] ??
anonymous
  • anonymous
Yes! So, putting it onto our graph now, |dw:1441202964806:dw|
anonymous
  • anonymous
yes okay
anonymous
  • anonymous
The third part just means that the angle used when graphing is \(\theta \le \frac{\pi}{3}\). We know that \(\theta = \tan^{-1}(\frac yx)\). So, we can try to get this into a form we're familiar with. \(\theta \le \frac\pi3\implies \tan^{-1}(\frac yx) \le \frac \pi3 \implies \frac yx \le \tan(\frac \pi3) \implies \frac yx \le \sqrt{3} \implies y\le x\sqrt{3}\)Now, we just need to graph this and shade in the right parts. Can you finish it up from there?
anonymous
  • anonymous
i just want to know what area to shade.. please help me with that
anonymous
  • anonymous
|dw:1441203537570:dw| Just test a point in each section to see whether it works or not. We need to satisfy \((x - 1)^2 + (y - \sqrt{3})^2 \le 1\) and \(y \le x\sqrt{3}\). Let's plug in a point above the line but within the circle, like (1, 2). Plugging it in, we can see that it satisfies the circle inequality, but not the line, so we know the shaded area is within the circle. Let's try plugging in a point in the circle, but below the line, like (1, 1). If we plug this in, we can see that it satisfies both inequalities, so we shade that region in.|dw:1441203715797:dw|

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