Answer:
C. [tex]z = -20\sqrt{3}-i\,20[/tex]
Step-by-step explanation:
The rectangular form of a complex number is represented by the following formula:
[tex]z = a+i\,b[/tex] (1)
Where each coefficient can be determined as function of the polar components:
[tex]a = r\cdot \cos \theta[/tex] (2)
[tex]b = r\cdot \sin \theta[/tex] (3)
Where:
[tex]r[/tex] - Magnitude of the complex number, dimensionless.
[tex]\theta[/tex] - Direction of the complex number, measured in radians.
If we know that [tex]r = 40[/tex] and [tex]\theta = \frac{7\pi}{6}[/tex], then the rectangular form of the number is:
[tex]a = 40\cdot \cos \frac{7\pi}{6}[/tex]
[tex]a = -20\sqrt{3}[/tex]
[tex]b = 40\cdot \sin \frac{7\pi}{6}[/tex]
[tex]b = -20[/tex]
The rectangular form of [tex]z=40\cdot\left(\cos \frac{7\pi}{6}+i\,\sin \frac{7\pi}{6}\right)[/tex] is [tex]z = -20\sqrt{3}-i\,20[/tex]. The correct answer is C.
