Which expression is the is a fourth root of -1+i sqrt3
1. the fourth root of 2 (cos(280 degrees)+i sin(280 degrees))
2.the fourth root of 2 (cos(30 degrees)+i sin(30 degrees))
3.the fourth root of 2 (cos(60 degrees)+i sin(60 degrees))
4.the fourth root of 2 (cos(320 degrees)+i sin(320 degrees))
Well, well, well! It seems like we're dealing with some complex numbers here. Let's break it down and have some fun!
First off, we need to find the fourth root of -1+i√3. Let's start with the given expressions:
1. We have the fourth root of 2, and in the form of cos(280 degrees) + i sin(280 degrees). Well, that's an interesting choice. But unfortunately, 280 degrees doesn't quite cut it for our little math adventure.
2. Ahh, now we have the fourth root of 2, but this time in the form of cos(30 degrees) + i sin(30 degrees). Now we're cooking! But let's not rush into anything just yet.
3. Ooh, look at this! We have the fourth root of 2, in the form of cos(60 degrees) + i sin(60 degrees). That sounds like a tempting option. But let's keep our options open.
4. Ah, the fourth root of 2 again, this time in the form of cos(320 degrees) + i sin(320 degrees). Interesting choice, my friend. But let's not run away with it just yet.
Now, to find the correct answer, we need to think about the angles that will work. Cosine and sine functions have a nice periodicity, repeating every 360 degrees. So, let's take a look at the options:
Option 1: 280 degrees
Option 2: 30 degrees
Option 3: 60 degrees
Option 4: 320 degrees
Now, if we sum up any of these angles, we want to end up with a multiple of 90 degrees. So, let's crunch some numbers!
280 degrees + 280 degrees = 560 degrees, which is not a multiple of 90 degrees.
30 degrees + 30 degrees = 60 degrees, which is also not a multiple of 90 degrees.
60 degrees + 60 degrees = 120 degrees, still not a multiple of 90 degrees.
320 degrees + 320 degrees = 640 degrees, nope, still not a multiple of 90 degrees.
Well, it looks like none of these options are giving us a multiple of 90 degrees. So, alas, the answer is none of the above! Time to go back to the drawing board and try again. Keep those mathematical gears turning, my friend!
To find the fourth root of -1 + i√3, we can use De Moivre's Theorem.
De Moivre's Theorem states that for any complex number z = r(cosθ + isinθ) raised to the power of n, the result can be found using the formula z^n = r^n(cos(nθ) + isin(nθ)).
In this case, we have z = -1 + i√3, which can be written in trigonometric form as z = 2(cos(120 degrees) + isin(120 degrees)). Note that √3 can be simplified as 2sin(60 degrees).
Now, to find the fourth root of z, we can use the formula z^(1/4) = 2^(1/4)(cos(120 degrees/4) + isin(120 degrees/4)).
Simplifying further, we have z^(1/4) = 2^(1/4)(cos(30 degrees) + isin(30 degrees)).
Therefore, the correct expression for the fourth root of -1 + i√3 is:
2^(1/4)(cos(30 degrees) + isin(30 degrees)).
Option 2: the fourth root of 2 (cos(30 degrees) + isin(30 degrees)) is the correct expression.
To solve this question, we need to find the fourth root of -1 + i√3, which is equivalent to finding the number raised to the power of 1/4 that gives us -1 + i√3.
Let's rewrite -1 + i√3 in polar form. The polar form of a complex number is given by r(cosθ + i sinθ), where r is the magnitude of the complex number and θ is its angle relative to the positive real axis.
The magnitude of -1 + i√3 is given by the formula |z| = √(Re(z)^2 + Im(z)^2), where Re(z) represents the real part of the complex number and Im(z) represents the imaginary part. Therefore, |z| = √((-1)^2 + (√3)^2) = √(1 + 3) = 2.
To find the angle θ, we can use the formula θ = arctan(Im(z) / Re(z)). In this case, θ = arctan(√3 / -1). Since √3 is positive and -1 is negative, the angle θ lies in the second quadrant (between 90 and 180 degrees). Using the arctan function, we get θ = 120 degrees.
Now, let's find the fourth root of -1 + i√3:
The fourth root of a complex number can be found by taking the fourth root of the magnitude and dividing the angle by 4.
1. The fourth root of 2 (cos(120 degrees) + i sin(120 degrees)):
- The magnitude of the fourth root of 2 is √2.
- Dividing the angle 120 degrees by 4 gives us 30 degrees.
- Therefore, the expression becomes √2 (cos(30 degrees) + i sin(30 degrees)).
2. The answer is the fourth root of 2 (cos(30 degrees) + i sin(30 degrees)).
r e^iT = r cos T + i r sin T
r cos T = -1
r sin T = sqrt 3
30, 60 90 triangle in second quadrant r = 2, T = 180-60 = 120
so
2e^120i
foruth root is 2^.25 e^(120/4)i
2^.25 e^30 i
2^.25 ( cos 30 + i sin 30)