Find the modulus and argument of the complex number ` 3sqrt2 -3sqrt2i` .

Explanation

Let ` z =3sqrt 2 - 3sqrt2i`
Let `r cos theta = 3sqrt2 and r sin theta = - 3 sqrt 2`
Squaring and adding , we get
` r^(2)cos^(2)theta +r^(2)sin^(2)theta = ( 3sqrt2)^(2) + ( -3sqrt2)^(2)`
` Rightarrow r^(2)(cos^(2)theta + sin^(2)theta) 18+18`
`Rightarrow r^(2) = 36`
` Rightarrow r =6 `
Now , ` cos theta = (3sqrt2)/r and sin theta = ( -3sqrt2)/r`
` Rightarrow costheta = ( 3sqrt2)/6 and sin theta = ( -3sqrt2)/6`
` Rightarrow cos theta = 1/sqrt2 and sin theta = ( -1)/sqrt2` ltbRgt Since the point representing the complex number ` 3sqrt2 -3 sqrt2i` lies in `4^("th")` quadrant.
` theta = pi/4 `
Thus the modulus and argument of the complex number ` 3sqrt2 -3sqrt2i " are" 6 and - pi/4` respectively .