find the cube roots of -216
answer in polar form
I did this already, but left a few steps for you to finish. Check the previous post of the same question
To find the cube roots of -216 in polar form, we can use the formula for the cube roots of a complex number.
Step 1: Express -216 in polar form.
To do this, we need to convert -216 to its polar form, which is given by:
-216 = 216(cos(π) + sin(π))
Step 2: Find the principal cube root.
The principal cube root of a complex number is given by:
z^(1/3) = r^(1/3) * [cos((θ + 2kπ)/3) + sin((θ + 2kπ)/3)]
where r is the modulus of the complex number, and θ is the argument of the complex number.
For -216, we have:
r = 216
θ = π
k = 0 (since we only need the principal root)
Substituting the values in the formula, we get:
(-216)^(1/3) = 216^(1/3) * [cos((π + 2(0)π)/3) + sin((π + 2(0)π)/3)]
= 6 * [cos(π/3) + sin(π/3)]
Step 3: Simplify the expression.
Using the values of cos(π/3) and sin(π/3) (which are 1/2 and √3/2 respectively), we simplify further:
(-216)^(1/3) = 6 * (1/2 + √3/2)
= 3 + 3√3 * i
So, the cube roots of -216 in polar form are:
3 + 3√3 * i
To find the cube roots of -216, we need to convert it to polar form and then find its cube root.
Step 1: Convert -216 to polar form:
To convert a complex number to polar form, we need to express it in the form r(cosθ + isinθ), where r represents the magnitude of the number and θ represents the angle it makes with the positive real axis.
For -216, we can calculate r by taking the magnitude of -216, which is √((-216)^2) = √46656 = 216.
To find θ, we need to consider the quadrant in which -216 lies. Since it is a negative real number, it lies in the third quadrant. We can calculate θ using the arctangent function: θ = atan(-216 / 0) = atan(-∞) = π.
So, -216 in polar form is 216(cosπ + isinπ).
Step 2: Find the cube root of -216 in polar form:
To find the cube root of a complex number in polar form, we need to take the cube root of its magnitude and divide the angle by 3.
Taking the cube root of 216, we get ∛216 = 6.
Dividing the angle π by 3, we get π/3.
Therefore, the cube roots of -216 in polar form are:
1. 6(cos(π/3) + isin(π/3))
2. 6(cos(π + π/3) + isin(π + π/3))
3. 6(cos(π + 2π/3) + isin(π + 2π/3))
Simplifying each expression will give you the exact polar form of the cube roots of -216.