convert root 3 cis (4pi/3) in to rectangular form
To convert the expression `√3cis(4π/3)` into rectangular form, we can use Euler's formula which states that `cis(x) = cos(x) + i*sin(x)`.
So, `√3cis(4π/3)` can be written as:
√3 * cos(4π/3) + i * √3 * sin(4π/3)
Now, we need to evaluate the cosine and sine of 4π/3.
cos(4π/3) = cos(240°) = -1/2
sin(4π/3) = sin(240°) = -√3/2
Now, we can substitute the values into the rectangular form expression:
√3 * (-1/2) + i * √3 * (-√3/2)
Simplifying:
-√3/2 - √3/2 * i
Therefore, the rectangular form of `√3cis(4π/3)` is `-√3/2 - √3/2 * i`.
To convert the expression `root 3 cis (4pi/3)` into rectangular form, we can use Euler's formula, which states that `e^(ix) = cos(x) + i * sin(x)`.
In this case, we have `cis(4pi/3)`, which is equivalent to `cos(4pi/3) + i * sin(4pi/3)`.
Now, let's evaluate `cos(4pi/3)` and `sin(4pi/3)`:
`cos(4pi/3) = -1/2`
`sin(4pi/3) = -sqrt(3)/2`
Putting it together, we get:
`root 3 cis (4pi/3) = root 3 * (-1/2 + i * (-sqrt(3)/2))`
Simplifying further:
`root 3 * (-1/2 + i * (-sqrt(3)/2)) = -root 3/2 - (root 3/2) * i * sqrt(3)`
Therefore, the rectangular form of `root 3 cis (4pi/3)` is `-root 3/2 - (root 3/2) * i * sqrt(3)`.
To convert a complex number from polar form to rectangular form, we can use the following formulas:
Real part (x) = r * cos(theta)
Imaginary part (y) = r * sin(theta)
Here, r represents the magnitude or length of the complex number, and theta represents the angle in radians.
In this case, we have the complex number root 3 cis (4pi/3), which can be written as:
r = root 3
theta = 4pi/3
Let's calculate the rectangular form step-by-step.
Step 1: Calculate the real part (x):
x = r * cos(theta)
= root 3 * cos(4pi/3)
Step 2: Calculate the imaginary part (y):
y = r * sin(theta)
= root 3 * sin(4pi/3)
Now, let's evaluate these calculations.
Step 1: Calculate x:
x = root 3 * cos(4pi/3)
= root 3 * (-1/2)
= -root 3 / 2
Step 2: Calculate y:
y = root 3 * sin(4pi/3)
= root 3 * √3/2
= 3√3 / 2
Therefore, the rectangular form of root 3 cis (4pi/3) is:
- root 3 / 2 + (3√3 / 2)i