![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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