Polar form





                     Polar form of complex numbers is another way of representing complex number.

                  z= a+ib is the rectangular form of a complex number.

                  z = r(cos θ + i sin θ) is the polar form of the complex number.

Here 'r' represents the absolute value and 'θ' represents argument of the complex number.

Representation:

    In the above diagram x- axis is the real axis and y-axis is the imaginary axis.  We will express real and imaginary part of the complex number in the form of r and θ where r is the modulus and θ is the argument of the complex number. θ is the angle made by the complex number with the real ( x) axis.

   In the above diagram, we know that

              r² = a² + b²

   By the trigonometric ratios we know that

             cos θ = a/r and sin θ = b/r.

   we can rewrite it as

           r cosθ = a  and  r sinθ = b.

  Substituting the value of a and b in the rectangular coordinate form we get

         z = r cosθ + i r sinθ

        z  = r(cosθ + i sinθ)

where r = ∣z∣ = √(a² + b²) and

          θ= tan ⁻¹ (b/a)

Note:  If a > 0 then θ= tan ⁻¹ (b/a).

           If  a < 0 then θ= tan ⁻¹ (b/a) + π

                                           or

                                 θ= tan ⁻¹ (b/a) + 180°          

Example:

Express z= 1+i in the form of polar.

Solution:

        Let us find the value of r.

        In the given complex number a = 1 and b = 1

       So the value of r is  √(a² + b²)

                                 r = √(1² + 1²)

                                    = √2

        and cos θ = a/r and sin θ = b/r.

               cos θ = 1/√2   and sin θ = 1/√2

       which implies θ = tan ⁻¹ (b/a)

                                   = tan ⁻¹ (1/1).

                                   =  tan ⁻¹ (1).

                                   =  π/4.

   So the polar form is z = 1/√2(cos π/4 + i sin π/4).

Example:

Convert in to rectangular form.

                   z = 2(cos 30° + i sin 30° )

Solution:

                   z  = 2 cos30°  + i 2sin 30°

              Here

                  a  = r cosθ and b = r sin θ.

            So a  = 2 cos 30° and b = 2 sin 30°

                 a  =  2(0.87)    and b = 2(0.5)

                 a  =  1.74        and b = 1

          The rectangular form is z = 1.74 + i


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