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
Rudiy27
3 years ago
11

Find the fourth roots of 16(cos 200° + i sin 200°).

Mathematics
1 answer:
NeTakaya3 years ago
3 0

Answer:

<em>See below.</em>

Step-by-step explanation:

To find roots of an equation, we use this formula:

z^{\frac{1}{n}}=r^{\frac{1}{n}}(cos(\frac{\theta}{n}+\frac{2k\pi}{n} )+\mathfrak{i}(sin(\frac{\theta}{n}+\frac{2k\pi}{n})), where k = 0, 1, 2, 3... (n = root; equal to n - 1; dependent on the amount of roots needed - 0 is included).

In this case, n = 4.

Therefore, we adjust the polar equation we are given and modify it to be solved for the roots.

Part 2: Solving for root #1

To solve for root #1, make k = 0 and substitute all values into the equation. On the second step, convert the measure in degrees to the measure in radians by multiplying the degrees measurement by \frac{\pi}{180} and simplify.

z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(0)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(0)\pi}{4}))

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{\pi}{4}))

z^{\frac{1}{4}} = 2(sin(\frac{5\pi}{18}+\frac{\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{\pi}{4}))

<u>Root #1:</u>

\large\boxed{z^\frac{1}{4}=2(cos(\frac{19\pi}{36}))+\mathfrack{i}(sin(\frac{19\pi}{38}))}

Part 3: Solving for root #2

To solve for root #2, follow the same simplifying steps above but change <em>k</em>  to k = 1.

z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(1)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(1)\pi}{4}))

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{2\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{2\pi}{4}))\\

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{\pi}{2}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{\pi}{2}))\\

<u>Root #2:</u>

\large\boxed{z^{\frac{1}{4}}=2(cos(\frac{7\pi}{9}))+\mathfrak{i}(sin(\frac{7\pi}{9}))}

Part 4: Solving for root #3

To solve for root #3, follow the same simplifying steps above but change <em>k</em> to k = 2.

z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(2)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(2)\pi}{4}))

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{4\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{4\pi}{4}))\\

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\pi))+\mathfrak{i}(sin(\frac{5\pi}{18}+\pi))\\

<u>Root #3</u>:

\large\boxed{z^{\frac{1}{4}}=2(cos(\frac{23\pi}{18}))+\mathfrak{i}(sin(\frac{23\pi}{18}))}

Part 4: Solving for root #4

To solve for root #4, follow the same simplifying steps above but change <em>k</em> to k = 3.

z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(3)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(3)\pi}{4}))

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{6\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{6\pi}{4}))\\

z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{3\pi}{2}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{3\pi}{2}))\\

<u>Root #4</u>:

\large\boxed{z^{\frac{1}{4}}=2(cos(\frac{16\pi}{9}))+\mathfrak{i}(sin(\frac{16\pi}{19}))}

The fourth roots of <em>16(cos 200° + i(sin 200°) </em>are listed above.

You might be interested in
Hey y’all I don’t understand, can you help me pls ASAP and show steps if u could
Shkiper50 [21]

Answer:

(x, y) = (-2, -6)

Step-by-step explanation:

Learn more at brainly.com/question/15168004

5 0
3 years ago
mario collected 3g seashells. jasmine collected 10 more seashells that mario. write the number of seashells they collected toget
Rufina [12.5K]

Answer:

Total seashells = (6g + 10) seashells.

Step-by-step explanation:

Given that Mario collected 3g seashells.

Given that Jasmine collected 10 more seashells that mario. It means she collected (3g+10) seashells.

It says to write the number of seashells they collected together in terms of g.

Total seashells = Mario's seashells + Jasmine's seashells.

Total seashells = 3g + (3g+10).

Total seashells = 6g + 10.

Hence, Total seashells = (6g + 10) seashells.

5 0
2 years ago
Compare 2/5 and 1/10. what is the denominator
omeli [17]
1/5 five being the denominator, because in math you must always simplify
7 0
3 years ago
Read 2 more answers
A standard-size bass drum has a diameter of 22 inches and is 18 inches deep. Find the volume of this drum
Lena [83]

Answer:

V=1,244.071

Step-by-step explanation:

I put the answer in decimal form and rounded up from .0706, but pi form in included in the pic. I hope this helped! Please leave Brainliest if it did and is right.

4 0
3 years ago
What is the solution set for 2x + 2 = 6, given the replacement set {1, 2, 3, 4}?
leva [86]
2x+2=6
2x=4
x=2

Hope this helps :)
7 0
3 years ago
Read 2 more answers
Other questions:
  • What is the product of 5/12 * 8/13 in lowest terms
    5·1 answer
  • Would 50% of a negative number be greater in value or less in value than the number itself? Give an example as part of your expl
    15·2 answers
  • How do I solve this step by step?
    12·1 answer
  • Please help me is trigonometry or geometry idk
    15·2 answers
  • <img src="https://tex.z-dn.net/?f=%28%20-%208x%7D%5E%7B2%7D%20%20-%205x%20-%206%29%20%2B%20%28%20-%20x%20%7B%20%7D%5E%7B2%7D%20-
    12·1 answer
  • Gwen purchased a fruit salad for $12 and two watermelons to cut up and add to fruit salad. The watermelons cost the same amount
    13·1 answer
  • Find the rule and the graph of the function whose graph can be obtained by performing the translation 4 units right and 3
    9·2 answers
  • What is the value of x when 3x &lt; -2x + 15
    15·2 answers
  • EASY MATH<br> Find all the zeros of f(x)= x^3 − 6x^2 + 13x − 20 given that 1−2i is a zero.<br> x=
    13·1 answer
  • Line passes through the point (8,4) and a slope of 5/4. Write equation in slope-intercept
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!