Express your answer in the form found using Euler's Formula, The cube roots of 2 + 3i
let z = 2+3i = √13(cos 56.3099..° + i sin 56.309.°) or √13cis 56.3099°
primary cube root
= z^(1/3) = (√13)^(1/3)cis 18.7699..° = (√13)^(1/3) e^(i 18.7699..°)
two more cube roots would be found by adding 360/3° or 120°
which would be (√13)^(1/3) e^(i 138.7699..)
and (√13)^(1/3) e^(i 256.7699...)
btw, another form of (√13)^(1/3) e^(i 18.7699..°) = 1.4519 + i(.4934)
I found this using degrees, I you want radians your result would be
(√13)^(1/3) e^(i .3276) , add 2π/3 to the argument to get the other two.
To express the cube roots of 2 + 3i using Euler's formula, we need to represent the complex number 2 + 3i in its polar form.
Let's first find the magnitude (r) and argument (θ) of the complex number:
Magnitude (r):
|r| = √((2^2) + (3^2))
|r| = √(4 + 9)
|r| = √13
Argument (θ):
θ = tan^(-1)((3/2))
θ ≈ 0.9828 radians
Now, using Euler's formula, we can express the complex number 2 + 3i as:
2 + 3i = √13 * e^(i * 0.9828)
To find the cube roots, we divide the exponent by 3:
(2 + 3i)^(1/3) = (√13 * e^(i * 0.9828))^(1/3)
Using the property of exponents, we can rewrite the cube root as:
(2 + 3i)^(1/3) = √13^(1/3) * e^((i * 0.9828) / 3)
Evaluating the cube root of √13:
√13^(1/3) ≈ 2.0801
Finally, simplifying the exponent:
(i * 0.9828) / 3 ≈ 0.3276i
Therefore, the expression of the cube roots of 2 + 3i using Euler's formula is:
(2 + 3i)^(1/3) ≈ 2.0801 * e^(0.3276i)
To express the cube roots of 2 + 3i using Euler's formula, we start by converting the complex number from rectangular form to polar form.
The rectangular form of the complex number is 2 + 3i, which means it can be represented as (2, 3) in terms of its real and imaginary components.
To find the polar form, we need to calculate the magnitude (r) and the angle (θ) of the complex number.
The magnitude (r) of a complex number can be found using the formula:
r = √(a^2 + b^2)
For the complex number 2 + 3i:
r = √(2^2 + 3^2) = √(4 + 9) = √13
The angle (θ) of a complex number can be found using the formula:
θ = arctan(b/a)
For the complex number 2 + 3i:
θ = arctan(3/2)
Now that we have the polar form of the complex number as (r, θ) = (√13, arctan(3/2)), we can express it using Euler's formula:
e^(iθ) = cos(θ) + i*sin(θ)
Substituting the values for θ, we get:
e^(i * arctan(3/2)) = cos(arctan(3/2)) + i*sin(arctan(3/2))
This is the expression of the cube roots of 2 + 3i using Euler's formula.