If z = 2cis 60°, find z^4 in rectangular form.
To find z^4 in rectangular form, we first need to understand what z represents.
In this case, z is a complex number represented in polar form, where 2 is the magnitude (or modulus) of the complex number, and 60° is the argument (or angle) of the complex number.
To find z^4, we need to raise z to the power of 4.
Step 1: Find the magnitude of z^4.
The magnitude of z^4 is obtained by raising the magnitude of z (2) to the power of 4:
|z^4| = |2|^4 = 2^4 = 16.
Step 2: Find the argument of z^4.
The argument of z^4 is obtained by multiplying the argument of z (60°) by 4:
arg(z^4) = 60° * 4 = 240°.
Step 3: Convert the magnitude and argument back into rectangular form.
To convert the magnitude and argument back into rectangular form, we can use the following formula:
x = r * cos(θ)
y = r * sin(θ)
where x and y represent the real and imaginary parts of the complex number, and r and θ represent the magnitude and argument, respectively.
Applying this formula, we get:
x = 16 * cos(240°)
y = 16 * sin(240°)
Step 4: Calculate x and y.
Using a calculator, we can evaluate the values of x and y as follows:
x = 16 * cos(240°) ≈ -8
y = 16 * sin(240°) ≈ -13.856
Therefore, z^4 in rectangular form is approximately -8 - 13.856i.
To find z^4 in rectangular form, we can first express z in rectangular form and then raise it to the power of 4.
Given z = 2cis(60°), we need to convert it into rectangular form. Recall that the cis function is defined as cis(θ) = cos(θ) + i*sin(θ).
Therefore, z = 2(cos(60°) + i*sin(60°)).
Next, let's expand z^4:
z^4 = [2(cos(60°) + i*sin(60°))]^4.
To simplify this expression, we can use De Moivre's theorem, which states that (cos θ + i*sin θ)^n = cos(nθ) + i*sin(nθ).
Using De Moivre's theorem, we can rewrite z^4 as:
z^4 = 2^4 * (cos(4*60°) + i*sin(4*60°)).
Simplifying the angles:
z^4 = 16 * (cos(240°) + i*sin(240°)).
Next, we evaluate the trigonometric functions for 240°:
cos(240°) = -1/2
sin(240°) = -√3/2
Substituting these values back into the expression for z^4, we get:
z^4 = 16 * (-1/2 + i*(-√3/2)).
Finally, simplifying, we have:
z^4 = -8 + 8i√3.
Therefore, z^4 in rectangular form is -8 + 8i√3.