Write the following complex number in polar form: z = 4 − 4i
Z = 4 - 4i.
tanAr = -4/4 = -1
Ar = -45o. = Reference angle.
A = -45 + 360 = 315o.
Z = X/cosA = 4/cos315 = 5.66 @ 315o or
(5.66,315).
NOTE: 315o = 7pi/4 Radians.
To convert the complex number z = 4 - 4i into polar form, we need to find the magnitude (r) and the angle (θ) of the complex number.
First, let's calculate the magnitude (r) of the complex number. The magnitude can be found using the Pythagorean theorem, which states that for a complex number a + bi, the magnitude is given by sqrt(a^2 + b^2).
In this case, a = 4 and b = -4.
So, r = sqrt(4^2 + (-4)^2) = sqrt(16 + 16) = sqrt(32) = √(16 * 2) = 4√2.
Next, we need to calculate the angle (θ) of the complex number. The angle can be found using the arctangent function, which is given by arctan(b/a).
In this case, a = 4 and b = -4.
So, θ = arctan((-4)/4) = arctan(-1) = -π/4.
Now, we have the magnitude (r) and the angle (θ) of the complex number z. In polar form, z can be represented as z = r(cosθ + isinθ).
Substituting the values we calculated, z = 4√2(cos(-π/4) + isin(-π/4)).
Therefore, the polar form of z = 4 - 4i is z = 4√2 * e^(-π/4)i, where e is Euler's number.