![Which Number Represents A Square Root Of 3 [cos(pi/2) + I Sin (pi/2)]?](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/MqWXL71zCVe43DIpTiqE5ufnn3R25BHp.png)
Answers
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Explanation:
We use De Moivre's Theorem. It says that if you started with
[tex]z = r*\left[\cos\left(\theta\right)+i*\sin\left(\theta\right)\right][/tex]
then raising both sides to the nth power leads to
[tex]z^n = r^n*\left[\cos\left(n*\theta\right)+i*\sin\left(n*\theta\right)\right][/tex]
The term out front r is raised to the nth power. We also have copies of n inside the trig functions.
The square root operation undoes the squaring operation, so we'll be dealing with n = 2.
If you were to go through each answer choice, then you'll find that squaring choice B will lead to the original expression given.
[tex]z = \sqrt{3}*\left[\cos\left(\frac{\pi}{4}\right)+i*\sin\left(\frac{\pi}{4}\right)\right]\\\\\\z^2 = (\sqrt{3})^2*\left[\cos\left(2*\frac{\pi}{4}\right)+i*\sin\left(2*\frac{\pi}{4}\right)\right]\\\\\\z^2 = 3*\left[\cos\left(\frac{\pi}{2}\right)+i*\sin\left(\frac{\pi}{2}\right)\right]\\\\\\[/tex]
Note that working this process in reverse means we apply the square root and we'll go from the expression in choice B to the original expression again.
The complex number Z = √3[cosπ/4 + i sinπ/4] represents the complex number √(3[cosπ/2 + i sinπ/2]) after using the De Moivre's theorem option (B) is correct.
What is a complex number?It is defined as the number which can be written as x+iy where x is the real number or real part of the complex number and y is the imaginary part of the complex number and i is the iota which is nothing but a square root of -1.
We have a complex number in the trigonometric form:
Z = √(3[cosπ/2 + i sinπ/2])
It is required to find the square root of the complex number.
To find the square root of the complex number we can use the De Moivre's theorem:
If
Z = r[cosθ + i sinθ]
Zⁿ = rⁿ[cosnθ + i sinnθ]
Using the above theorem:
Z = √3[cos[(1/2)π/2] + i sin[(1/2)π/2]]
Z = √3[cosπ/4 + i sinπ/4]
Thus, the complex number Z = √3[cosπ/4 + i sinπ/4] represents the complex number √(3[cosπ/2 + i sinπ/2]) after using the De Moivre's theorem option (B) is correct.
Learn more about the complex number here:
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Related Questions
In general, as sample size increases: a.the standard error will remain the same regardless of changes in sample size. b.the standard error fluctuates in size. c.the standard error increases in size. d.the standard error is a constant. e.the standard error decreases in size.
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Answer:
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Step-by-step explanation:
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Answers
Answer:
x=7
Step-by-step explanation:
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Answers
Answer:
$180
Step-by-step explanation:
simple interest means that interest is paid every year on the original $200
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$200(3)=$60
Since the interest is the same, 3(60) = $180
Hope this helps
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Type your equation in the form y=mx+b. Do not use any spaces in your response.
Answers
Answer:
y = 5x - 32
Step-by-step explanation:
The equation of a line in slope- intercept form is
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y = 5x - 2 ← is in slope- intercept form
with slope m = 5
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Answers
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Answers
Answer:
690
Step-by-step explanation:
0.69 x 10^3=
0.69 x 1000=
690
Answer:
Step-by-step explanation:
0.69*1000= 690
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Answer:
true
Step-by-step explanation:
because i just finished doing this in geometry
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Answer:
i believe the answer is 13 ?
Step-by-step explanation:
1300 divided by 100 is 13 im not sure if this is right tho sry if im wrong.
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Answer:
1) No it repeating
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3) yes this terminates
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
If x = 5y - 1 and 3x + 5y = -43, what is the value of y?
Answers
Answer:
y is the value of 1
Step-by-step explanation:
no need for one
Answer:
y = - 2
Step-by-step explanation:
Substitute x = 5y - 1 into the other equation
3(5y - 1) + 5y = - 43 ← distribute parenthesis and simplify left side
15y - 3 + 5y = - 43
20y - 3 = - 43 ( add 3 to both sides )
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y = - 2
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Answer:
D) -[tex]\frac{3}{2}[/tex]
Step-by-step explanation:
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Answer:
D,_3/2
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Answers
Answer:
No
Step-by-step explanation:
a^2+b^2=c^2
5^2= 25
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25+49=74
square root of 74 does not equal 9
Answer:
No.
Step-by-step explanation:
Pythagorean Theorem: a²+b²=c²
5²+7²=9²
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Answers
The expression that represents the area of the gray region, the area of the land remaining in the backyard is 3x²+10x-17 square yards. The correct option is B.
The area of the gray region is the area of the rectangular backyard minus the area of the circular pond.
The area of the rectangular backyard is given as 6x+14x-12 square yards, which simplifies to 20x-12 square yards.
The area of the circular pond is given as 3x²-4x+5 square yards.
So, the area of the gray region is:
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sin(45)=13/x
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Answer:
number 1 matches 8/12, number 2 matches 11/12, number 3 matches 12/24, number 4 matches 11/24, number 5 matches 10/12
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Answer:
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Answer:
Step-by-step explanation:
Simple interest is the agreed upon amount that was first deposited. Even if the depositor leaves his interest in the account, the amount paid is on the account is on the 5000 dollars, not the 5100 it will be at the end of the second year.
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among 240 students it was fount three - fourth of the students preferred to play hockey and one-eighth of the students preferred to play cricket the number of students whole prefer to play hockey and cricket respectively are.
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Answers
Answer:
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Step-by-step explanation:
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Answers
Answer:
f
and
c
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Step-by-step explanation:
f and c
E and C
Step-by-step explanation:
E on the front
C on the top
I hope it helps
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Answer
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Step-by-step explanation:
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Answers
Answer:
[tex]\frac{104}{100}[/tex]
Step-by-step explanation:
You would multiply 104/100 to the price because it would be 100%, (basically the price), plus the 4/100, the tax.
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Answers
Answer:
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Step-by-step explanation: