17) Convert the rectangular form of the complex number into polar form.
z= 2-2√3i
A: z=4(cos 2pi/3 + i sin 2pi/3)
B: z = 4cis 5pi/3
C: z = 2 √2 (cos 11pi/6 + i sin 11 pi/6)
D: z = 2 √2cis 5pi/6
18) Find the quotient:
z₁= 10 (cos 9pi/8+ i sin 9pi/8) and z₂ = 2 (cos pi/4 + i sin pi/4)
A: z₁/z₂ = 20 cis (pi)
B: z₁/z₂ = 8 cis (11pi/8)
C: z₁/z₂= 5 cis (7pi/8)
D: z₁/z₂ = 12cos (5pi/4)
Solution:
17) We can first find the magnitude or modulus of the complex number using the formula:
|z| = sqrt(a^2 + b^2)
where a and b are the real and imaginary parts of the complex number, respectively.
In this case, a = 2 and b = -2√3, so
|z| = sqrt(2^2 + (-2√3)^2) = sqrt(16) = 4.
Next, we can find the argument or angle of the complex number using the formula:
arg(z) = atan(b/a)
where atan is the inverse tangent function.
In this case, arg(z) = atan(-2√3/2) = atan(-√3) = -pi/3 (since tan(-pi/3) = -√3).
However, we need the angle to be between 0 and 2π, so we add 2π to get arg(z) = 5π/3.
Therefore, the polar form of the complex number is:
z = 4 cis (5π/3)
which is equivalent to option B.
18) To find the quotient of two complex numbers in polar form, we divide their moduli and subtract their arguments:
z₁/z₂ = |z₁|/|z₂| * cis (arg(z₁) - arg(z₂))
In this case, we have:
|z₁| = 10 and |z₂| = 2
arg(z₁) = 9π/8 and arg(z₂) = π/4
Substituting into the formula, we get:
z₁/z₂ = 10/2 * cis (9π/8 - π/4) = 5 cis (7π/8)
which is equivalent to option C.