Answer:
[tex]8.48(\cos (45)-i\sin (45))[/tex]
Step-by-step explanation:
We are given the complex number, [tex]z=6-6i[/tex].
Now, the trigonometric form of a complex number [tex]z=a+ib[/tex] is given by, [tex]|z|(\cos \theta+i\sin \theta)[/tex].
Since, we have that, [tex]|z|=\sqrt{a^{2}+b^{2}}[/tex]
So, [tex]|z|=\sqrt{6^{2}+(-6)^{2}}[/tex]
i.e. [tex]|z|=\sqrt{36+36}[/tex]
i.e. [tex]|z|=\sqrt{72}[/tex]
i.e. [tex]|z|=8.48[/tex]
Further, we have that, [tex]\theta = \arctan(\frac{b}{a})[/tex]
So, [tex]\theta = \arctan(\frac{-6}{6})[/tex]
i.e. [tex]\theta = \arctan(-1)[/tex]
i.e. θ = - 45°
So, the trigonometric form of the given complex number is,
[tex]8.48(\cos (-45)+i\sin (-45))[/tex] i.e. [tex]8.48(\cos (45)-i\sin (45))[/tex]
Hence, the trigonometric form is [tex]8.48(\cos (45)-i\sin (45))[/tex].