demoivre theorem
(2sqrt3+2i)^5
r=(2sqrt3+2)^2
r^2 = (2√3)^2 + 2^2 = 16, so r=4
(2√3 + 2i) = (4,π/6)
(2√3 + 2i)^5 = (4^5,5π/6) = (-512√3,512)
whats (2√3)^2 equal to?
To use the De Moivre's theorem, you first need to understand the polar form of a complex number. In polar form, a complex number is represented as r(cosθ + isinθ), where r is the modulus (distance from the origin to the point representing the complex number) and θ is the argument (angle between the positive x-axis and the line connecting the origin and the point).
Now, let's go step by step to find the value of (2√3 + 2i)^5 using the De Moivre's theorem.
Step 1: Convert the complex number to polar form.
To find the modulus (r), we use the formula r = √(a^2 + b^2), where a is the real part of the complex number and b is the imaginary part.
Given the complex number (2√3 + 2i), a = 2√3 and b = 2.
So, r = √((2√3)^2 + 2^2) = √(12 + 4) = √16 = 4.
To find the argument (θ), we use the formula θ = arctan(b/a).
θ = arctan(2/(2√3)).
Step 2: Simplify the argument.
To simplify the argument, we can rationalize the denominator.
θ = arctan(2/(2√3)) * √3/√3.
θ = arctan(2√3/6).
Since arctan(√3) = π/3, we have:
θ = arctan(√3/3) = π/6.
Step 3: Apply De Moivre's theorem.
De Moivre's theorem states that (r(cosθ + isinθ))^n = r^n (cos(nθ) + isin(nθ)).
Applying De Moivre's theorem, we have:
(2√3 + 2i)^5 = 4^5 (cos(5(π/6)) + isin(5(π/6))).
Simplifying:
4^5 = 1024.
cos(5(π/6)) = cos(5π/6) = cos (5π) / cos (6π) = -1/2.
sin(5(π/6)) = sin(5π/6) = sin (5π) / cos (6π) = √3/2.
(2√3 + 2i)^5 = 1024 (-1/2 + i√3/2).
Thus, (2√3 + 2i)^5 = -512 + 512√3i.