if z1= 3+3i and z2=7(cos(5pi/9) + i sin (5pi/9)), then z1/z2= blank

Answers

Answer 1

[tex]z1=\stackrel{a}{3}+\stackrel{b}{3}i~~ \begin{cases} r = \sqrt{a^2+b^2}\\ r = \sqrt{18}\\[-0.5em] \hrulefill\\ \theta =\tan^{-1}\left( \frac{b}{a} \right)\\ \theta =\frac{\pi }{4} \end{cases}~\hfill z1=\sqrt{18}\left[\cos\left( \frac{\pi }{4} \right) i\sin\left( \frac{\pi }{4} \right) \right] \\\\[-0.35em] ~\dotfill[/tex]

[tex]\cfrac{z1}{z2}\implies \cfrac{\sqrt{18}\left[\cos\left( \frac{\pi }{4} \right) i\sin\left( \frac{\pi }{4} \right) \right]} {7\left[\cos\left( \frac{5\pi }{9} \right) i\sin\left( \frac{5\pi }{9} \right) \right]} \\\\[-0.35em] ~\dotfill\\\\ \qquad \textit{division of two complex numbers} \\\\ \cfrac{r_1[\cos(\alpha)+i\sin(\alpha)]}{r_2[\cos(\beta)+i\sin(\beta)]}\implies \cfrac{r_1}{r_2}[\cos(\alpha - \beta)+i\sin(\alpha - \beta)] \\\\[-0.35em] ~\dotfill[/tex]

[tex]\cfrac{z1}{z2}\implies \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{\pi }{4}-\frac{5\pi }{9} \right)+i\sin\left( \frac{\pi }{4}-\frac{5\pi }{9} \right) \right] \\\\\\ \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{-11\pi }{36} \right) +i\sin\left( \frac{-11\pi }{36} \right) \right]\implies \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{83\pi }{36} \right) +i\sin\left( \frac{83\pi }{36} \right) \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \cfrac{z1}{z2}\approx 0.348~~ + ~~0.496i~\hfill[/tex]

Answer 2

The value of z1/z2 is √18/ 7 (cos ( 11π/36 ) - isin ( 11π/36 )).

What is complex number?

"A complex number is the sum of a real number and an imaginary number and it is of the form x + iy and is usually represented by z".

For the given situation,

z1= 3+3i and

z2= 7(cos(5π/9) + i sin (5π/9))

To divide the complex numbers, both should be in same form.

Convert z1 in polar form.

z is of the form x+iy, so, r=[tex]\sqrt{x^{2}+y^{2} }[/tex]

⇒[tex]r=\sqrt{3^{2}+3^{2} }[/tex]

⇒[tex]r=\sqrt{18}[/tex]

θ = [tex]tan^{-1}(\frac{b}{a} )[/tex]

⇒[tex]tan^{-1}(\frac{3}{3} )[/tex]

⇒[tex]tan^{-1}(1)[/tex]

⇒[tex]45[/tex]°

The polar form is of the form, z= r (cosθ + i sinθ),

⇒ z1 = [tex]\sqrt{18}[/tex] (cosπ/4 + isinπ/4)

The formula for dividing complex number is

z1/z2 = r1(cos θ1 + isin θ1) / r2(cos θ2 + isin θ2)

⇒ z1/z2 = r(cosθ + isinθ)

where, r = r1/r2 and θ = (θ1 - θ2)

z1/z2 = [tex]\sqrt{18}[/tex] (cos π/4 + isin π/4) / 7 (cos 5π/9 + isin 5π/9)

⇒ r = [tex]\sqrt{18}[/tex] / 7 and

θ = (π/4 - 5π/9 )

⇒ θ = (-11π/36)

cos(-θ) = cos θ

⇒cos( -11π/36 ) = cos ( 11π/36 )

sin(-θ) = -sin θ

⇒ sin ( -11π/36 ) = -sin ( 11π/36 )

Thus, z1/z2 = [tex]\sqrt{18}[/tex] / 7 (cos ( 11π/36 ) - isin ( 11π/36 ))

Hence we can conclude that the value of z1/z2 is

√18/ 7 (cos ( 11π/36 ) - isin ( 11π/36 )).

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Related Questions

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To construct angle MNO congruent to the given angle ABC, the following conditions must be satisfied:

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Answer:

option B is correct answer. ∆SVZ~=∆YVX, SSS

Step-by-step explanation:

reasons,

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VX=VZ, S

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Given that there are three triangles and these three triangles are connected to each other at the centres' A, B, and C.

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Answer:

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Step-by-step explanation:

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I hope this helps!

1/3(6-9j)

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Answer:

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Answer:

1: 2.8

Step-by-step explanation:

15:42

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1: 2.8

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Step-by-step-explanation:

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Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

I hope this helps. Also, there's this free app called Quick Math. If you need the steps, you should check there first, and maybe you should try that for other math problems as well.

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Answer:

7

Step-by-step explanation:

7

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Answer:

Sample Response: Since the statement is true, there are an infinite number of solutions to the system. This statement also indicates that the lines are equivalent.

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Answer:

G

Step-by-step explanation:

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Choose the best answer below concerning the confidence interval for the difference between the true proportions of adult females minus the true proportion of adult males who use 2 or more credit/debit cards.

Answers

Using the z-distribution, it is found that the correct option is:

[tex]p_1 - p_2 \pm z\sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}[/tex]

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[tex]p \pm zs[/tex]

In which:

p is the estimate of the proportion.z is the critical value.s is the standard error.

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A proportion [tex]p_1[/tex] in a sample of size [tex]n_1[/tex].[tex]p_2[/tex] in a sample of size [tex]n_2[/tex].

Hence, the standard errors are:

[tex]s_1 = \sqrt{\frac{p_1(1 - p_1)}{n_1}}, s_2 = \sqrt{\frac{p_2(1 - p_2)}{n_2}}[/tex]

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[tex]p = p_1 - p_2[/tex]

[tex]s = \sqrt{s_1^2 + s_2^2} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}[/tex]

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[tex]p_1 - p_2 \pm z\sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}[/tex]

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Step-by-step explanation:

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Answer:

Step-by-step explanation:

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Answer:

  2√2

Step-by-step explanation:

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z = 2

Step-by-step explanation:

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Answer:

x = 90y = 43

Step-by-step explanation:

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______

Hope it helps ⚜

Step-by-step explanation:

6 tickets=$96

1 ticket =96/6

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

State the postulate or theorem you would use to prove each pair of triangles congruent.
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d) AAS
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Answers

Answer:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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How do you need help?
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See image.

Step-by-step explanation:

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Answers

Answer:

isoselses!

Step-by-step explanation:

i hope that helps

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