How do I write 5e^(-2j) in real and imaginary forms?
To write the complex number 5e^(-2j) in real and imaginary forms, we can use Euler's formula. Euler's formula states that for any real number x, e^(ix) can be expressed as cos(x) + isin(x), where i is the imaginary unit.
First, let's break down the complex number 5e^(-2j):
5 is the magnitude or absolute value of the complex number.
e^(-2j) is the exponential part of the complex number.
To write it in real and imaginary forms, we can separate it as follows:
Real Part: The real part is obtained by multiplying the magnitude with the cosine of the exponent.
Real Part = 5 * cos(-2)
Imaginary Part: The imaginary part is obtained by multiplying the magnitude with the sine of the exponent.
Imaginary Part = 5 * sin(-2)
Now let's calculate these values.
Using the trigonometric identities, we have:
cos(-2) = cos(2) [cosine is an even function]
And
sin(-2) = -sin(2) [sine is an odd function]
Since cos(2) ≈ -0.416 and sin(2) ≈ 0.909, we calculate:
Real Part ≈ 5 * (-0.416) ≈ -2.080
Imaginary Part ≈ 5 * (-0.909) ≈ -4.545
Therefore, 5e^(-2j) can be written in real and imaginary forms as approximately -2.080 - 4.545j.