Write the complex number in

Question

Standard a+bi form

 

4(cos (2pi/3)+isin(2pi/3))

 

Polar form with the angle in radians

 

-4+2i

Answer 1

To get from polar form to standard form, all you need to do is calculate directly. There aren't any tricks to it. First note that

 

cos(2pi/3)=-1/2   and    sin(2pi/3)=sqrt(3)/2

 

So     4(cos(2pi/3)+isin(2pi/3)) = 4(-1/2+i*sqrt(3)/2)

      = -2 + i*2sqrt(3)

 

Notice that we can write the "i" before or after 2sqrt(3) since complex multiplication is commutative.

 

 

To get from standard form to polar form, we need to do a little work before. We want to know the magnitude r and direction Θ of our complex vector. Those are the variables requisite in polar form, and

 

-4+2i = r(cosΘ+isinΘ)

 

To get the magnitude, we calculate r = sqrt((-4)²/+(2)²) = sqrt(20) = 2sqrt(5)

 

To get the direction, we calculate Θ = tan^-1(2/(-4)) = tan^-1(-1/2) ≈ 2.678

 

So -4+2i = 2sqrt(5)*[cos(2.678)+isin(2.678)]