Answer :

Given Complex number is Z = (i^{25})^{3}

⇒ Z = i^{75}

⇒ Z = i^{74}.i

⇒ Z = (i^{2})^{37}.i

We know that i^{2} = - 1

⇒ Z = ( - 1)^{37}.i

⇒ Z = ( - 1).i

⇒ Z = - i

⇒ Z = 0 - 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 =

θ = arg(z) = argument of complex number =

Now for the given problem,

⇒

⇒

⇒

⇒ |z| = 1

⇒

Since x>0,y<0 complex number lies in 4^{th} quadrant and the value of θ will be as follows - 90^{0}≤θ≤0^{0}.

⇒

⇒ .

⇒

⇒

∴ The Polar form of Z = (i^{25})^{3} is .

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