If rectangular form of complex number is shown as z=52+532i then its polar form is represented as – 

  1. 5(cos(2π3)isin(2π3))
  2. 5(cos(π3)isin(π3))
  3. 5(cos(2π3)+isin(2π3))
  4. 5(cos(π3)+isin(π3))

Answer (Detailed Solution Below)

Option 4 : 5(cos(π3)+isin(π3))
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Detailed Solution

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CONCEPT:

Point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P.

If P represent the nonzero complex number z = x + iy.

Here r=x2+y2=|z| is called modulus of given complex number.

The argument of Z is measured from positive x-axis only.

Let z = r (cosθ + i sinθ) is polar form of any complex number then following ways are used while writing θ for different quadrants –

For first quadrant, θ=tan1yx

For second quadrant θ=πtan1yx

For third quadrant θ=π+tan1yx

For fourth quadrant θ=tan1yx

CALCULATION:

Given complex number is z=52+532i

rcosθ=52,rsinθ=532

By squaring and adding, we get –

r2(cos2θ+sin2θ)=1004=25

∴ r = 5

cosθ=52r=525=12 and sinθ=532r=5325=32

Since it is in first quadrant, θ=π3

So, on comparing with z = r (cosθ + i sinθ), we can write as 5(cos(π3)+isin(π3)) 

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