ANSWER
[tex]2 \sqrt{2} ( \cos( \frac{7\pi}{4} + i \sin( \frac{7\pi}{4} ) ) [/tex]
EXPLANATION
The complex number shown has coordinates (2,-2)
or
[tex]z = 2 - 2i[/tex]
The modulus is
[tex] |z| = \sqrt{ {2}^{2} + {( - 2)}^{2} } [/tex]
[tex]|z| = \sqrt{ 8} = 2 \sqrt{2} [/tex]
The argument is
[tex] \theta= \tan^{ - 1} ( \frac{ - 2}{2} )[/tex]
[tex] \theta= \tan^{ - 1} ( -1) = \frac{7\pi}{4} [/tex]
The polar form is
[tex]2 \sqrt{2} ( \cos( \frac{7\pi}{4} + i \sin( \frac{7\pi}{4} ) ) [/tex]
The first option is correct.
