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4-1.Complex numbers
medium
निम्नलिखित सम्मिश्र संख्याओं का मापांक एवं कोणांक ज्ञात कीजिए।
$\frac{1}{1+i}$
Option A
Option B
Option C
Option D
Solution
We have $\frac{1}{1+i}=\frac{1-i}{(1+i)(1-i)}=\frac{1-i}{1+1}=\frac{1}{2}-\frac{i}{2}$
Let $\frac{1}{2}=r \cos \theta,-\frac{1}{2}=r \sin \theta$
Proceeding as in part $(i)$ above, we get $r=\frac{1}{\sqrt{2}} ; \cos \theta=\frac{1}{\sqrt{2}}, \sin \theta=\frac{-1}{\sqrt{2}}$
Therefore $\theta=\frac{-\pi}{4}$
Hence, the modulus of $\frac{1}{1+i}$ is $\frac{1}{\sqrt{2}},$ argument is $\frac{-\pi}{4}$.
Standard 11
Mathematics
