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Find the modulus and argument of the following complex numbers and hence express each of them in the polar form : √3 + i

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Given complex number is Z=\(\sqrt{3}\)+i 

We know that the polar form of a complex number Z=x+iy is given by Z=|Z|(cosθ+isinθ) 

Where, 

|Z|=modulus of complex number= \(\sqrt{x^2+y^2}\)

θ =arg(z)=argument of complex number= tan-1 \(\Big(\frac{|y|}{|x|}\Big)\)

Now for the given problem,

⇒ |z| = \(\sqrt{(\sqrt{3})^2+(1)^2}\) 

⇒ |z| = \(\sqrt{3+1}\) 

⇒ |z| = \(\sqrt{4}\)

⇒ θ = tan-1\(\Big(\frac{1}{\sqrt{3}}\Big)\) 

Since x>0,y>0 complex number lies in 1st quadrant and the value of θ will be as follows 0°≤θ≤90°.

⇒ θ = \(\frac{\pi}{6}\)

⇒  z = \(\sqrt{2}\Big(cos\Big(\frac{\pi}{6}\Big) + isin\Big(\frac{\pi}{6}\Big)\Big)\) 

∴ The Polar form of Z=\(\sqrt{3}\)+i is z =\({2}\Big(cos\Big(\frac{\pi}{6}\Big) + isin\Big(\frac{\pi}{6}\Big)\Big)\) 

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