1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
geniusboy [140]
3 years ago
8

Find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of -9i.

Mathematics
1 answer:
vitfil [10]3 years ago
6 0

Answer:

Step-by-step explanation:

Given z = x+iy

The polar form of a complex number z = r(cos\theta + isin\theta)

The nth root of the complex number is expressed according to de moivre's  theorem as;

z^{\frac{1}{n} }  = [r(cos\theta + isin\theta)]^{\frac{1}{n} } \\z^{\frac{1}{n} } = \sqrt[n]{r} (cos(\frac{\theta+2nk}{n} ) + isin(\frac{\theta+2nk}{n}))

r is the modulus of the complex  number and \theta is the argument

r = √x²+y²

\theta = tan^{-1} \frac{y}{x}

Given z = -9i

r = √0+(-9)²

r = √81

r = 9

\theta = tan^{-1}\frac{-9}{0}  \\\theta = tan^{-1}-\infty\\\theta = -90^{0}

The argument will be equivalent to 180-90 = 90°

The forth root of -9i will be expressed as shown according to de moivre's theorem;

z_k^{\frac{1}{4} } = \sqrt[4]{9}  (cos(\frac{90+2(4)k}{4} ) + isin(\frac{90+2(4)k}{4}))\\z_k^{\frac{1}{4} } = \sqrt[4]{9}  (cos(\frac{90+8k}{4} ) + isin(\frac{90+8k}{4}))\\

The complex roots are at when k = 0, 1, 2 and 3

When k = 0;

z_0 = \sqrt[4]{9}  (cos(\frac{90}{4} ) + isin(\frac{90}{4}))\\z_0 = \sqrt[4]{9}  (cos(23) + isin(23))\\\\when\ k =1\\z_1 = \sqrt[4]{9}  (cos(\frac{90+8}{4} ) + isin(\frac{90+8}{4}))\\z_1 = \sqrt[4]{9}  (cos(25 ) + isin(25))\\\\when\ k =2\\z_2 = \sqrt[4]{9}  (cos(\frac{90+16}{4} ) + isin(\frac{90+16}{4}))\\z_2 = \sqrt[4]{9}  (cos(27 ) + isin(27))\\\\when\ k =3\\z_3 = \sqrt[4]{9}  (cos(\frac{90+24}{4} ) + isin(\frac{90+24}{4}))\\z_3 = \sqrt[4]{9}  (cos(29 ) + isin(29))\\

Note that all the degrees are rounded to the nearest whole number.

You might be interested in
a car detailing company charges $30 plus $18 per hour. Another company charges $25 plus $20 per hour. What is a reasonable domai
Oxana [17]

Answer:

What is a reasonable domain for the two functions?

x > 0


5 0
3 years ago
Someone please help !!!!
AnnZ [28]
It is A. when you synthetic divide you end up underneath 2 7 0. the 0 is the remainder. the other makes 2x+7
6 0
3 years ago
Read 2 more answers
Batman slings a 320 ft. climbing cable from a helicopter to his batmobile below. If the cable tightens up
Leokris [45]

Answer:

188 ft

Step-by-step explanation:

         helicopter

___________________________________________

               |\   54°

               |    \

               |       \    320 ft

               |          \

               |      54°  \

-------------+--------------+-------------------------------

                        x

cos 54° = x/(320 ft)

x = 320 ft × cos 54°

x = 188 ft

Answer: 188 ft

5 0
2 years ago
12. Write the equation of a parabola with vertex at<br> (2, -3) and focus at (2,5),
Tatiana [17]

Step-by-step explanation:

The basic form of equation:

(x-h)²=4a(y-k),

(h,k)=coordinates of vertex

(h, k+a) = coordinate of focus

For given parabola:

axis of symmetry: x=2

(h, k) =(2,-3)

(h, k+a)=(2,5)

k+a=5

-3+a=5

a=8(distance from vertex to focus on the axis of symmetry)

equation: (x-2)²=4×8(y+3)

(x-2)²= 32(y+3)

7 0
3 years ago
At a local clothing store, the price of an $80 winter coat was reduced by 25% when spring arrived. Once it was summer time, the
Oksanka [162]

Answer:45$

Step-by-step explanation:

6 0
3 years ago
Other questions:
  • X to the power of 6 minus 2 to the power of 6
    14·1 answer
  • Which body part is the same length as the persons length?
    13·2 answers
  • How many 1/8 pound servings of walnuts, s , are in 3/4 pound of walnuts?
    6·2 answers
  • A Student select a marble from a bag, keeps it and select another. The bag contains 5 Green marbles 4 black marbles and 2 blue m
    15·1 answer
  • ( √2 - √3 )² =<br> a. 5 - 2√6<br> b. 5 - √6<br> c. 1 - 2√6<br> d. 1 - √2<br> e. 1
    13·1 answer
  • What is the equation of the line in slope-intercept form?
    13·1 answer
  • Please help me on this question.
    13·2 answers
  • If RT= 19 meters, find LW.
    5·1 answer
  • Find the circumference of a circle with a radius of 18.59 inches to the nearest tenth of an inch.
    12·1 answer
  • How much greater is 8 than –3?
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!