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