Question

Convert (3√2 + 3√2i) from rectangular coordinates to polar coordinates. Express the angle in radians within the interval (0 ≤ θ < 2Π/), with (r ≥ 0).

a) (3√2)
b) (6√2)
c) (3√2 + Π/4)
d) (6√2 + Π/4)

Answer 1

The complex number (3√2 + 3√2i) in polar coordinates is (6√2, π/4), with the magnitude r being 6 and the angle θ being π/4 radians.

Step-by-step explanation:

The student has asked to convert the complex number (3√2 + 3√2i) from rectangular coordinates to polar coordinates and to express the angle in radians. To do this, we use the polar form r(cos(θ) + i sin(θ)), where r is the magnitude of the complex number and θ is the angle with the positive part of the real axis.

First, we need to find r, which is the magnitude of the vector. r = √(Re(z)^2 + Im(z)^2), where Re(z) and Im(z) are the real and imaginary parts of z, respectively. Substituting the values, we get r = √((3√2)^2 + (3√2)^2) = √(18 + 18) = √36 = 6.

Next, we find the angle θ, which is θ = tan^-1(Im(z)/Re(z)). Since the real and imaginary components are equal, θ = tan^-1(1) = π/4. Because the complex number (3√2 + 3√2i) is in the first quadrant, this is our angle in radians.

Combining these values, the polar coordinates are (6√2, π/4). The correct answer is d) (6√2, π/4).