How do you convert $- 4 + 8 i$ $-4 + 8i$ to polar form?

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How do you convert $- 4 + 8 i$ $-4 + 8i$ to polar form?

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Answer 1 · 2017-07-15T11:11:57

In polar coordinates, the point will be $(r, θ) = (6.92 i, - 1.11 i)$ $(r,theta)=(6.92i, -1.11i)$

Explanation:

For coordinates in polar form, $(r, θ)$ $(r, theta)$ , we need to find the value of $r$ $r$ and the value of $θ$ $theta$ .

We will use Pythagoras' theorem to find $r$ $r$ and the $tan$ $tan$ function to find $θ$ $theta$ .

$r = \sqrt{{(- 4)}^{2} + 8 i^{2}} = \sqrt{16 - 64} = \sqrt{- 48} = 6.92 i$ $r=sqrt((-4)^2+8i^2)=sqrt(16-64) =sqrt(-48)=6.92i$

(since $8 i^{2} = 8 i \times 8 i = 8 \sqrt{- 1} \times 8 \sqrt{- 1} = 64 \times - 1 = - 64$ $8i^2=8ixx8i=8sqrt(-1)xx8sqrt(-1)=64xx-1=-64$ )

$tan θ = \frac{o p p}{a d j} = \frac{8 i}{- 4} = - \frac{8}{4} i = - 2 i$ $tantheta=(opp)/(adj)=(8i)/-4=-8/4i=-2i$

So $θ = {tan}^{- 1} (- 2 i) = - 1.11 i$ $theta=tan^-1(-2i)=-1.11i$

Answer 2 · 2017-07-15T12:31:14

$(4 \sqrt{5}, 2.03)$ $(4sqrt5,2.03)$

Explanation:

$to convert from cartesian to polar form$ $"to convert from "color(blue)"cartesian to polar form"$

$that is (x, y) \to (r, θ) where$ $"that is "(x,y)to(r,theta)" where"$

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

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

$here x = - 4 and y = 8$ $"here " x=-4" and "y=8$

$\Rightarrow r = \sqrt{{(- 4)}^{2} + 8^{2}} = \sqrt{80} = 4 \sqrt{5}$ $rArrr=sqrt((-4)^2+8^2)=sqrt80=4sqrt5$

$- 4 + 8 i is in the second quadrant$ $-4+8i" is in the second quadrant "$

$so we must ensure that θ is in the second quadrant$ $"so we must ensure that "theta" is in the second quadrant"$

$θ = {tan}^{- 1} (2) = 1.11 \leftarrow related acute angle$ $theta=tan^-1(2)=1.11larrcolor(red)" related acute angle"$

$\Rightarrow θ = (π - 1.11) = 2.03 \leftarrow in second quadrant$ $rArrtheta=(pi-1.11)=2.03larrcolor(red)" in second quadrant"$

$\Rightarrow - 4 + 8 i \to (- 4, 8) \to (4 \sqrt{5}, 2.03)$ $rArr-4+8ito(-4,8)to(4sqrt5,2.03)$

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How do you convert $- 4 + 8 i$ $-4 + 8i$ to polar form?

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How do you convert −4+8i -4 + 8i to polar form?

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