Final answer:
The complex number 8 e⁵ i π/4 in Cartesian form is 4√2 + 4√2 i, and the complex number 3 - 3 √3 i in polar form is (6, -π/3).
Step-by-step explanation:
To write the complex number 8 e⁵ i π/4 in Cartesian form, we can use Euler's formula, which states that eiθ = cos(θ) + i sin(θ). Therefore, the Cartesian form of the given complex number is obtained by evaluating cos(π/4) and sin(π/4). Both of these trigonometric functions have the value of √2/2 when θ is π/4, so:
8 e⁵ i π/4 = 8(cos(π/4) + i sin(π/4)) = 8(√2/2 + i √2/2) = 4√2 + 4√2 i.
For the complex number 3 - 3 √3 i in polar form, we calculate the magnitude and the angle. The magnitude (r) is the square root of the sum of the squares of the real and imaginary parts:
r = √(3² + (-3√3)²) = √(9 + 27) = √36 = 6
The angle (θ), also known as the argument, can be found by taking the arctangent of the imaginary part divided by the real part:
θ = atan2(-3√3, 3) = -π/3, because atan2 is used to take into account the signs of both components for the correct quadrant.
The polar form of 3 - 3 √3 i is then (6, -π/3).