Can someone please give me a step by step answer on how to find the complex cube roots of 8(cos(4π/5)+isin(4π/5))? I need to know how to do this for my trig final next week.
We will use De Moivre's theorem
8(cos(4π/5) + isin(4π/5)^(1/3)
= 8^(1/3) (cos (1/3)(4π/5) + i sin (1/3)(4π/5) )
= 2(cos (4π/15) + i sin(4π/15) )
for a youtube of De Moivre's theorem check out this:
http://www.youtube.com/watch?v=NrwWv9JHdI4
(rcisθ)^n = r^n cis(nθ)
cuberoot = 1/3 power, so
∛8(cos(4π/5)+isin(4π/5))
= ∛8 cis((4π/5)/3)
= 2cis(4π/15)
Now, since cisθ = cis(θ+2nπ)
cis 4π/5 = cis 14π/5 = cis 24π/5
(2cis 4π/15)^3 = 8cis 4π/5
(2cis (4π/15 + 2π/3))^3 = (2cis 14π/15)^3 = 8cis 14π/5
(2cis (4π/15 + 4π/3))^3 = (2cis 24π/15)^3 = 8cis 24π/5
So, you need to evaluate sin & cos of 4π/15,14π/15,24π/15
adding another 2π takes us around to 34π/15 = 2π + 4π/15 and we're back where we started.
Sure, I can definitely help you with finding the complex cube roots of the given number.
Step 1: Write the given number in polar form
The number is expressed in the form 8(cos(4π/5) + isin(4π/5)). This is already in polar form with magnitude 8 and angle 4π/5.
Step 2: Find the cube root of the magnitude
To find the cube root of 8, we can take the cube root of its magnitude. So, the magnitude of each of the complex cube roots will be ∛8 = 2.
Step 3: Find the angle of each complex cube root
To find the angles of the complex cube roots, we divide the angle of the given number by 3 since we are finding cube roots. In this case, the angle is 4π/5. So, to find the angles of the complex cube roots, we divide 4π/5 by 3:
(4π/5) / 3 = (4/5π) * (1/3) = (4/15)π
Step 4: Convert the polar form to rectangular form
Now we have the magnitude (2) and the angles (4/15)π for the complex cube roots. Using these values, we can convert each root into rectangular form.
The rectangular form of a complex number is given by z = a + bi, where a is the real part and b is the imaginary part.
For the first complex cube root:
The real part, a = magnitude * cos(angle) = 2 * cos((4/15)π)
The imaginary part, b = magnitude * sin(angle) = 2 * sin((4/15)π)
For the second complex cube root:
To find the second root, add 2π to the angle:
The real part, a = 2 * cos((4/15)π + 2π) = 2 * cos((14/15)π)
The imaginary part, b = 2 * sin((4/15)π + 2π) = 2 * sin((14/15)π)
For the third complex cube root:
To find the third root, add 4π to the angle:
The real part, a = 2 * cos((4/15)π + 4π) = 2 * cos((24/15)π)
The imaginary part, b = 2 * sin((4/15)π + 4π) = 2 * sin((24/15)π)
So, the three complex cube roots of 8(cos(4π/5) + isin(4π/5)) are:
1. 2(cos((4/15)π) + isin((4/15)π))
2. 2(cos((14/15)π) + isin((14/15)π))
3. 2(cos((24/15)π) + isin((24/15)π))
Please note that the angles are given in radians, and the fractions are simplified as much as possible.