Sure, let's rewrite the given complex number [tex]\( z = 4 \left( \cos \left( \frac{7\pi}{6} \right) + i \sin \left( \frac{7\pi}{6} \right) \right) \)[/tex] in rectangular form.
1. Recall the polar form expression of a complex number:
[tex]\[ z = r (\cos \theta + i \sin \theta) \][/tex]
2. Here, the magnitude [tex]\( r \)[/tex] is 4 and the angle [tex]\( \theta \)[/tex] is [tex]\( \frac{7\pi}{6} \)[/tex].
3. The real part of the complex number is given by:
[tex]\[ \text{Real Part} = r \cos \theta = 4 \cos \left( \frac{7\pi}{6} \right) \][/tex]
4. The imaginary part of the complex number is given by:
[tex]\[ \text{Imaginary Part} = r \sin \theta = 4 \sin \left( \frac{7\pi}{6} \right) \][/tex]
So, the rectangular form [tex]\( a + bi \)[/tex] of the complex number is composed of the calculated real part and imaginary part.
Therefore, the real part calculated is:
[tex]\[ -3.4641016151377553 \][/tex]
The imaginary part calculated is:
[tex]\[ -1.999999999999999 \][/tex]
Thus, combining both, the rectangular form of the complex number is:
[tex]\[ z = -3.4641016151377553 - 2i \][/tex]
(Note: The value -1.999999999999999 is extremely close to -2 and can be expressed as -2 for simplicity.)
Therefore, the final rectangular form is approximately:
[tex]\[ z = -\sqrt{12} - 2i \][/tex]
