pre calculus
posted by joe .
cube roots of 64i?

pre calculus 
MathMate
Convert the given number into the form
a+bi = α(cos(β)+i sin(β))
where
α = sqrt(a^2+b^2)
β = sin^{1}(b/α)
If the number is plot in the complex (Z) plane, it will be more evident.
For example, 64i will have
α=64 (the distance from origin)
β=90° (sin^{1}64/64=1)
The equivalent angles are:
90°
450°
810°
(we only need 3)
Divide each of these angles by , to get β1, β2, & β3. And take α'=α^(1/3)=64^(1/3)=4
The three cube roots are then
z1=α'(cos(β1)+isin(β1)
=4(cos(30°)+isin(30°))
z2=4(cos(150°)+isin(150°))
z3=4(cos(270°)+isin(270°))
Check by expanding z1^3, z2^3 and z3^3 to get back 64i. 
pre calculus typo 
MathMate
β=90° (sin164/64=sin^{1}1=π/2)
Respond to this Question
Similar Questions

Math
Roots Ok, what about roots? Roots of polynomials? 
precalculus
find the cube roots of 216 answer in polar form and complex 
precalculus
find the cube roots of 216 answer in polar form 
PreCalculus
Hello, I have tried solving this using the rational roots theorem and none of the roots seem to be working. I'm trying to figure out where I went wrong. 3x^3 2x^2  7x  4 = 0 
trig
what is the cube root of (64i)? 
precalc
express the roots of unity in standard form a+bi. 1.) cube roots of unity 2.) fourth roots of unity 3.) sixth roots of unity 4.) square roots of unity 
precalculus
express the roots of unity in standard form a+bi. 1.) cube roots of unity 2.) fourth roots of unity 3.) sixth roots of unity 4.) square roots of unity 
Pre calculus
For the equation 2x^25x^3+10=0 find the number the number of complex roots and the possible number of real roots. 
PreCalculus
Rewrite each quadratic equation in the form ax^2+bx+c=0. Then,use technology to solve each by graphing. ROund you answers to the nearest hundredth, where necessary. a) 3x^2+30 = 19x Answer: 3x^2+19x+30 Roots: x = 3 b) 6x^2= 25x24 … 
PreCalculus 11
What is the entire radial form of 3* cube root of 2?