To convert the polar form of a complex number into its rectangular form, we use the formula [tex]\( z = r \left( \cos \theta + i \sin \theta \right) \)[/tex], where [tex]\( r \)[/tex] is the magnitude and [tex]\( \theta \)[/tex] is the angle (in radians).
Given the polar form:
[tex]\[ z = 7 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) \][/tex]
We need to find the rectangular form, which involves calculating the real part and the imaginary part separately.
1. Calculate the real part:
The real part is given by [tex]\( r \cos \theta \)[/tex].
[tex]\[ \text{Real part} = 7 \cos \frac{\pi}{2} \][/tex]
2. Calculate the imaginary part:
The imaginary part is given by [tex]\( r \sin \theta \)[/tex].
[tex]\[ \text{Imaginary part} = 7 \sin \frac{\pi}{2} \][/tex]
Next, let's substitute the values of [tex]\( \cos \frac{\pi}{2} \)[/tex] and [tex]\( \sin \frac{\pi}{2} \)[/tex]:
[tex]\[
\cos \frac{\pi}{2} = 0
\][/tex]
[tex]\[
\sin \frac{\pi}{2} = 1
\][/tex]
Substituting these values back into the equations for the real and imaginary parts:
[tex]\[
\text{Real part} = 7 \times 0 = 0
\][/tex]
[tex]\[
\text{Imaginary part} = 7 \times 1 = 7
\][/tex]
Therefore, the rectangular form of the given complex number is:
[tex]\[
(0, 7)
\][/tex]
So, the correct answer is:
[tex]\[ B. (0, 7) \][/tex]
