Find the real and imaginary parts of the following complex numbers:
(i) 1.5e^jπ/4
(ii) je^1.2 + j^2 e
e^jπ/4 = cos(π/4) + j sin(π/4) = 1/√2 + 1/√2 j
Assuming j^2 = -1, you have
-e + e^1.2 j
To find the real and imaginary parts of a complex number, we can use Euler's formula, which states that for any complex number z = re^(jθ), the real part is given by r * cos(θ) and the imaginary part is given by r * sin(θ).
Let's find the real and imaginary parts of the given complex numbers:
(i) 1.5e^(jπ/4):
Here, r = 1.5 and θ = π/4.
The real part is: 1.5 * cos(π/4)
To evaluate this, we can use the value of cos(π/4) = sqrt(2)/2.
So, the real part is: 1.5 * sqrt(2)/2 = sqrt(2)/2 ≈ 1.06.
The imaginary part is: 1.5 * sin(π/4)
To evaluate this, we can use the value of sin(π/4) = sqrt(2)/2.
So, the imaginary part is: 1.5 * sqrt(2)/2 = sqrt(2)/2 ≈ 1.06.
Therefore, the real and imaginary parts of 1.5e^(jπ/4) are approximately equal to sqrt(2)/2.
(ii) je^1.2 + j^2e:
In this case, we have two terms: je^1.2 and j^2e.
For the first term, r = 1 and θ = 1.2.
The real part is: 1 * cos(1.2) ≈ 0.362.
The imaginary part is: 1 * sin(1.2) ≈ 0.932.
For the second term, j^2 = -1 (since j^2 = -1 in complex numbers).
The real part is: -1 * cos(0) = -1.
The imaginary part is: -1 * sin(0) = 0.
Adding these two terms together gives: 0.362 + (-1) + 0.932j
Simplifying, we get: -0.638 + 0.932j.
Therefore, the real part of je^1.2 + j^2e is approximately equal to -0.638, and the imaginary part is approximately equal to 0.932.