Question

5. ( 14 points) Write the complex number in trigonometric form. Begin by sketching the graph to help you find the argument 0. 3 - 31

Answer 1

$\blacksquare\,\,\underline{Convert\,\,Complex\,\, Number \,\,to\,\, Trigonometric\,\,Form}:$
2 . A -4-3-2-1 २ 4 (8 - 31) -3
$To\,\,convert\,\,from\,\,complex\,\, number\,\,to\,\, trigonometric\\form \,\,that\,\,is,\,\,\,\,\,\,\,x+iy \rightarrow\,r(cos\Theta +i\,sin\Theta) \,\,\,using\\\\\bullet \,\,\,\,r=\sqrt{x^2+y^2}\\\\\bullet\,\,\, \Theta=tna^{-1}(\frac{y}{x})\,\,\,;\,\,\,\,[-\pi<\Theta\leq \pi\,] \\\\Here\,\,x=3\,\,and\,\,y=-3\,\,,\\\\\Rightarrow\,\,\,\,r =\sqrt{3^2+(-3)^2}\,=\sqrt{9+9}\,=\sqrt{18}\,=3\sqrt{2}\\\\ By\,\,the\,\,diagram\,\,we\,\,can\,\,see\,\,3-3i\,\,is\,\,in\,\,fourth \,\, quadrant\\so\,\,we\,\,must\,\,ensure\,\,that\,\,\Theta \,\,is\,\,in\,\,the\,\, fourth\,\, quadrant\,.\\\\So,\,\,\,\,\Theta =tan^{-1}(\frac{-3}{3})=tan^{-1}(-1)=-\frac{\pi}{4}\\\\ Since,\,\,\Theta\,\,is\,\,in\,\,fourth\,\, quadrant,\,\,\,\Theta=-\frac{\pi}{4}\,\,. \\\\\therefore\,\,\,\,3-3i\,\,=\,2\sqrt{3}(cos(-\frac{\pi}{4})+i\,sin(-\frac{\pi}{4}) \\\\\Rightarrow\,\,3-3i\,\,=\,2\sqrt{3}(cos(\frac{\pi}{4})-i\,\,sin(\frac{\pi}{4}) \\\\This\,\,is\,\,the\,\, required\,\, trigonometric\,\,form\,.$

$[\,\,\underline{Note}:\hspace{10pt}Using\,\,the\,\,graph\,\,we\,\,have\,\,the\,\, argument \\\Theta\,= -45\degree=-\frac{\pi}{4}\,\,.\,\,]$

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