What is the rectangular form of the complex number z^4 if z = 2cis 60°?
To find the rectangular form of a complex number, we need to convert it from the trigonometric (polar) form to the rectangular form. The formula for converting polar coordinates (r, θ) to rectangular coordinates (x, y) is:
x = r * cos(θ)
y = r * sin(θ)
Given that z = 2cis 60°, we can see that the magnitude (r) is 2, and the angle (θ) is 60°.
To find z^4, we raise both the magnitude and the angle to the fourth power:
Magnitude: r^4 = 2^4 = 16
Angle: θ^4 = 60° * 4 = 240°
Now we can use the formula to convert these values to rectangular form:
x = r * cos(θ) = 16 * cos(240°)
y = r * sin(θ) = 16 * sin(240°)
Using a calculator, we can find the cos(240°) and sin(240°) values:
cos(240°) ≈ -0.5
sin(240°) ≈ -0.87
Plugging these values into the formulas:
x ≈ 16 * (-0.5) = -8
y ≈ 16 * (-0.87) = -13.92
Therefore, the rectangular form of z^4 is -8 - 13.92i.