Find the three cube roots of 216(cos 315 + I sin 315)
To find the cube roots of a complex number in polar form, we first need to convert the number to rectangular form.
Given number: 216(cos 315 + i sin 315)
The rectangular form of the number is:
216(cos 315) + i(216 sin 315)
=216(cos 315) + i(216(-√2/2))
= 216(-√2/2) + i(-108√2)
= -108√2 - 108√2i
Now we need to find the cube root of this number.
Let the cube root be x + yi.
(x + yi)^3 = -108√2 - 108√2i
x^3 + 3ix^2y - 3xy^2 - y^3i = -108√2 - 108√2i
(x^3 - 3xy^2) + (3x^2y - y^3)i = -108√2 - 108√2i
Since the real and imaginary parts must be equal, we have:
x^3 - 3xy^2 = -108√2
3x^2y - y^3 = -108√2
We will now solve these equations to find x and y, and hence the cube root of the number.