Question

The polar form of (i^41)^3

Answer 1

The polar form of the given complex number  (i⁴¹)³ would be given as z = cos(π/2) - isin(π/2).

Given:

The given complex number is equal to (i⁴¹)³.

To Find:

The polar form of the given complex number.

Solution:

→ The term 'i' is known as iota and its value is $\sqrt{-1}$.

           ∴ z = (i⁴¹)³ = i¹²³

           ∴ z = (i)¹²⁰ × (i)³

          ∴ z  = ($\boldsymbol{\sqrt{-1} }$ )¹²⁰ × (i)³

          ∴ z  = (($\boldsymbol{\sqrt{-1}}$)²)⁶⁰ × (i²)i

          ∴ z  = (1)⁶⁰ × (-1)i

          ∴ z  = -(i)

→ The polar form of a complex number 'z' is given r(cosθ + isinθ).

• 'r' is known as the amplitude of the complex number.
• Here 'θ' is known as the argument of the complex number.

→ For a complex number (z) = a + ib:

(i) The amplitude of the complex number would be equal to $\boldsymbol{\sqrt{a^{2} +b^{2} } }$ .

(ii) The argument of the complex number can be calculated as follows:

     $\boldsymbol{Firstly\:we\:calculate \:the\:value\:of\:'\alpha '\:given\:by:\:\alpha =tan^{-1}|\frac{b}{a} |}$

The argument of the complex number would be:

• 'α' if the complex number lies in the first quadrant.
• (π - α) if the complex number lies in the second quadrant.
• (α - π) if the complex number lies in the third quadrant.
• -α if the complex number lies in the fourth quadrant.

→ For the given complex number : z = -(i)

On comparing the given complex number with z = a + i(b), we get 'a' is equal to 0 and 'b' is equal to -1.

$\boldsymbol{\therefore\:Amplitude\:of\:the\:complex\:number(r)=\sqrt{a^{2} +b^{2} } }$

                                                                   $\boldsymbol{\therefore r=\sqrt{a^{2} +b^{2} } }\\\\\boldsymbol{\therefore r=\sqrt{0^{2} +(-1)^{2} } }\\\\\boldsymbol{\therefore r=\sqrt{0+1} }\\\\\boldsymbol{\therefore r=1 }$

→ The amplitude (r) of the complex number is equal to 1.

→ We will now calculate the argument of the complex number:

 $\boldsymbol{Firstly\:we\:calculate \:the\:value\:of\:'\alpha '\:given\:by:\:\alpha =tan^{-1}|\frac{b}{a} |}$

                              $\boldsymbol{\therefore \alpha =tan^{-1}|\frac{-1}{0} |=\pi /2}$

∵ The complex number lies on the negative y-axis.

∴ The argument (θ) of the complex number would be equal to -α.

∴ The argument (θ) of the complex number would be equal to $\boldsymbol{-\pi /2}$.

→ The polar form of  the complex no. 'z' would be equal to r(cosθ + isinθ).

Polar Form: z = r(cosθ + isinθ)

                ∴ z = 1 × [cos(-π/2) + isin(-π/2)]

                ∴ z = cos(π/2) - isin(π/2)

Hence the polar form of the given complex number  (i⁴¹)³ would be given as z = cos(π/2) - isin(π/2).

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