Could someone help me with this-we learned this today in school but I'm not getting it now that I'm home with my homework-I have lots of these to do can someone help with this one and then maybe it'll click-thank you
Find (sqrt3/2 -1/2i)^4 by using DeMoivre's Theorem
I have to convert to polar, use the theorem and put it back into rectangular form
tan(t) = y/x = (-1/2) /(sqrt(3)/2) = -1/sqrt(3)
so, t = -pi/6
r^2 = x^2 + y^2 = 3/4 + 1/4 = 1
r = 1
so, point z = (1,-pi/6)
z^4 = 1^4 cis 4*(-pi/6) = 4 cis -2pi/3
x = r cos -2pi/3 = 1 * -1/2
y = r sin -2pi/3 = 1 * -sqrt(3)/2
z^4 = (-1/2, -sqrt(3)/2)
Just for confirmation,
(sqrt(3)/2 - i/2)^4 =
9/16 - 4*3*sqrt(3)/(8*2)i + 6*3/4*1/4 i^2 - 4*sqrt(3)/2 * 1/8 i^3 + 1/16 i^4
= 9/16 - 3sqrt(3)/4 i - 18/16 + 1/4 sqrt(3) i + 1/16
-8/16 - sqrt(3)/2 i
Certainly! I can help you with that. First, we'll need to convert the given complex number into polar form.
To convert a complex number from rectangular form (a + bi) to polar form (r(cosθ + isinθ)), we use the formulas:
- r = √(a^2 + b^2), where a and b are the real and imaginary components of the number.
- θ = arctan(b/a), where a is the real component and b is the imaginary component.
In this case, we have (sqrt(3)/2 - (1/2)i.
Let's calculate the polar form of this complex number:
First, find the value of r:
r = √((sqrt(3)/2)^2 + (-1/2)^2)
r = √(3/4 + 1/4)
r = √(4/4)
r = √1
r = 1
Next, calculate the value of θ:
θ = arctan((-1/2)/(sqrt(3)/2))
θ = arctan(-1/sqrt(3))
θ = -π/3 (since tan(-π/3) = -1/√3)
Therefore, the polar form of (sqrt(3)/2 - (1/2)i) is:
1(cos(-π/3) + isin(-π/3))
Now, we can apply DeMoivre's Theorem, which states that (r(cosθ + isinθ))^n = r^n(cos(nθ) + isin(nθ)).
In this case, we need to find the fourth power of our complex number.
So, using DeMoivre's Theorem, we have:
(1(cos(-π/3) + isin(-π/3)))^4 = 1^4(cos(4(-π/3)) + isin(4(-π/3)))
Simplifying:
(cos(4(-π/3)) + isin(4(-π/3))) = cos(-4π/3) + isin(-4π/3)
= cos(-2π/3) + isin(-2π/3)
= -1/2 - (√3/2)i
Finally, we convert this back to rectangular form if required.
Therefore, (sqrt(3)/2 - (1/2)i)^4 in rectangular form is -1/2 - (√3/2)i.
I hope this helps! Let me know if you have any further questions.