How do you express the complex number in trigonometric form: 5-5i?

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Answer 1 · 2018-04-06T12:09:43

$5 \sqrt{2} (cos (- \frac{π}{4}) + i sin (- \frac{π}{4}))$ $5sqrt(2)(cos(-pi/4)+isin(-pi/4))$

To convert from $z = x + i y$ $z=x+iy$ form to $z = r (cos θ + i sin θ)$ $z=r(costheta+isintheta)$ form, you need to find $r$ $r$ and $θ$ $theta$ .

$r = \sqrt{x^{2} + y^{2}}$ $r=sqrt(x^2+y^2)$ and $tan θ = \frac{y}{x}$ $tantheta=y/x$

So, $r = \sqrt{5^{2} + {(- 5)}^{2}} = \sqrt{50} = 5 \sqrt{2}$ $r=sqrt(5^2+(-5)^2)=sqrt(50)=5sqrt(2)$

$tan θ = \frac{5}{- 5} \Rightarrow tan θ = - 1 \Rightarrow θ = - \frac{π}{4}$ $tantheta=5/-5=>tantheta=-1=>theta=-pi/4$

$therefore 5-5i=5sqrt(2)(cos(-pi/4)+isin(-pi/4))$

Answer 2 · 2018-04-06T12:12:07

In trigonometric form: $7.07(cos 45-isin 45)$

$Z= a+ib=5 - 5i$ . Modulus: $|Z|=sqrt(a^2+b^2)$

Modulus: $|Z|=sqrt(5^2+(-5)^2)$

$=sqrt 50 ~~ 7.07$ Argument: $tan alpha=|b/a|= |5/-5|=1$

$:.alpha =tan^-1 (1) = 45^0 ; Z$ lies on fourth

quadrant. $:. theta=360-alpha =315^0 or (-45)^0$

In trigonometric form expressed as

$Z=|Z|(cos theta+isin theta)$

$:. Z=7.07(cos 315+isin 315)$ or

$Z=7.07(cos 45-isin 45)$ [Ans]

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