Question

Find the modulus and the principal value of the argument of the number $1-i$

• A. $\sqrt{2},\pi /4$
• B. $\sqrt{2},-\pi /4$
• C. $\sqrt{2},-\pi /3$
• D. $\sqrt{2},3\pi /4$

Answer 1

The correct option is B $\sqrt{2},-\pi /4$

Let $1=r\cos \theta ,-1=r\sin \theta$

Squaring and adding these relations, we get $r^2({\cos }^2\theta +{\sin }^2\theta )=1^2+{(-1)}^2=2$.

Since, $co{s}^2\theta +si{n}^2\theta =1,\Rightarrow r=\pm \sqrt{2}$

$r$ cannot be negative

Hence. $r=\sqrt{2}$

Then $\cos \theta =\frac{1}{\sqrt{2}},\sin \theta =\frac{-1}{\sqrt{2}},$

The value of $\theta$ between-$\pi$ and $\pi$ which satisfies these equations is $-\left(\pi /4\right)$ Thus $|1-i|=r=\sqrt{2}$ and $arg(1-i)=-\left(\pi /4\right).$

Ans: B