What is the trigonometric form of $(4 - 2 i)$ $(4-2i)$ ?

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What is the trigonometric form of $(4 - 2 i)$ $(4-2i)$ ?

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Answer 1 · 2017-07-13T18:26:35

$2 \sqrt{5}$ $2sqrt5$ $c i s$ $cis$ $(- 0.46)$ $(-0.46)$

Explanation:

To find the trigonometric form, we have to know $r$ $r$ , the distance of the point from the origin, and $θ$ $theta$ , the angle.

We can use the following formulas:

$r = \sqrt{a^{2} + b^{2}}$ $r = sqrt(a^2+b^2)$

$tan θ = \frac{b}{a}$ $tan theta = b/a$

$r = \sqrt{4^{2} + {(- 2)}^{2}}$ $r = sqrt(4^2+(-2)^2)$
$r = \sqrt{20}$ $r = sqrt(20)$
$r = 2 \sqrt{5}$ $r = 2sqrt(5)$

$tan θ = - \frac{2}{4}$ $tan theta = -2/4$
$θ = {tan}^{- 1} (- \frac{2}{4})$ $theta = tan^-1(-2/4)$
$θ = - 0.46$ $theta = -0.46$

So, the trigonometric form is $2 \sqrt{5}$ $2sqrt5$ $c i s$ $cis$ $(- 0.46)$ $(-0.46)$ or $2 \sqrt{5} (cos (- 0.46) + i sin (- 0.46))$ $2sqrt5 (cos (-0.46) + i sin (-0.46))$ .

Answer 2 · 2017-07-13T19:13:41

$2 \sqrt{5} (cos (0.46) - i sin (0.46))$ $2sqrt5(cos(0.46)-isin(0.46))$

Explanation:

$to convert from cartesian to trig. form$ $"to convert from "color(blue)"cartesian to trig. form"$

$that is (x, y) \to r (cos θ + i sin θ) use$ $"that is " (x,y)tor(costheta+isintheta)" use"$

$∙ x r = \sqrt{x^{2} + y^{2}}$ $•color(white)(x)r=sqrt(x^2+y^2)$

$∙ x θ = {tan}^{- 1} (\frac{y}{x}) x; - π < θ \leq π$ $•color(white)(x)theta=tan^-1(y/x)color(white)(x);-pi < theta<=pi$

$here x = 4 and y = - 2$ $"here "x=4" and " y=-2$

$\Rightarrow r = \sqrt{4^{2} + {(- 2)}^{2}} = \sqrt{20} = 2 \sqrt{5}$ $rArrr=sqrt(4^2+(-2)^2)=sqrt20=2sqrt5$

4-2i is in the fourth quadrant so we must ensure that $θ$ $theta$ is in the fourth quadrant.

$θ = {tan}^{- 1} (\frac{1}{2}) = 0.46 \leftarrow related acute angle$ $theta=tan^-1(1/2)=0.46larrcolor(red)" related acute angle"$

$\Rightarrow θ = - 0.46 \leftarrow in fourth quadrant$ $rArrtheta=-0.46larrcolor(red)" in fourth quadrant"$

$\Rightarrow 4 - 2 i = 2 \sqrt{5} (cos (- 0.46) + i sin (- 0.46))$ $rArr4-2i=2sqrt5(cos(-0.46)+isin(-0.46))$

$\Rightarrow 4 - 2 i = 2 \sqrt{5} (cos (0.46) - i sin (0.46))$ $rArr4-2i=2sqrt5(cos(0.46)-isin(0.46))$

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What is the trigonometric form of $(4 - 2 i)$ $(4-2i)$ ?

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What is the trigonometric form of $(4 - 2 i)$ $(4-2i)$ ?