root 3 cis 4pi/3 turn it in to rectangular form
the answer is root3/2 and 3/2i explain
To convert the given expression in polar form, root 3 cis 4pi/3, to rectangular form, we can use Euler's formula:
e^(iθ) = cosθ + isinθ
Given: root 3 cis 4pi/3
We can rewrite it as:
root 3 * e^(i(4pi/3))
Using Euler's formula, we get:
root 3 * (cos(4pi/3) + isin(4pi/3))
cos(4pi/3) = -1/2
sin(4pi/3) = -sqrt(3)/2
Plugging these values back into the expression, we have:
root 3 * (-1/2 + i*(-sqrt(3)/2))
Simplifying, we get:
-root 3/2 - (sqrt(3)/2)i
Therefore, the rectangular form of root 3 cis 4pi/3 is -root 3/2 - (sqrt(3)/2)i.
To convert a complex number given in polar form, which consists of a magnitude (r) and an angle (θ), into rectangular form, you can use the following formulas:
Real part (x): r * cos(θ)
Imaginary part (y): r * sin(θ)
In this case, the complex number is represented as:
√3 * cis(4π/3)
The magnitude (r) is √3, and the angle (θ) is 4π/3.
Using the formulas mentioned earlier:
Real part (x) = √3 * cos(4π/3)
Imaginary part (y) = √3 * sin(4π/3)
Calculating the values:
Real part (x) = √3 * (-1/2) = -√3/2
Imaginary part (y) = √3 * (√3/2) = 3/2
Therefore, the rectangular form of the complex number √3 * cis(4π/3) is:
-√3/2 + (3/2)i