1.

The complex number (-1 - i) is expressed in polar form as1. (cos(5π/4) + i sin(5π/4))2. √2(cos(3π/4) - i sin(3π/4))3. √2(cos(π/4) + i sin(π/4))4. √2(cos(5π/4) - i sin(5π/4))

Answer» Correct Answer - Option 2 : √2(cos(3π/4) - i sin(3π/4))

Concept:

Consider a complex number z = a + ib,

Polar form is given by z = r(cosθ + isin θ), where

 r = \(\rm \sqrt {a^2 +b^2}\) and θ = tan-1 \(\left(\rm\frac{Im(z)}{Re(z)} \right)\)

Quadrant

Sign of x and y

Principle value of Argument

I

x > 0, y > 0

\(\rm \tan^{-1} \frac{y}{x}\)

II

x < 0, y > 0

π - \(\rm \tan^{-1} \left|\frac{y}{x}\right|\)

III

x < 0, y < 0

-π + \(\rm \tan^{-1} \left|\frac{y}{x}\right|\)

IV

x > 0, y < 0

\(-\rm \tan^{-1} \left|\frac{y}{x}\right|\)

 

Calculation:

Given complex number, -1 - i

|z| = r = \(\rm \sqrt {{(-1)}^2 +{(-1)}^2}\)

= √2

Now,   -1 - i can be represented as point P(-1, -1),which lies in 3rd quadrant.

Now, θ  = -π + tan-1\(\left(\rm\frac{Im(z)}{Re(z)} \right)\)

= -π + π/4 = -3π/4

∴ Polar form: z = √2[cos(-3π/4) + i sin(-3π/4)]

= √2[cos(3π/4) - i sin(3π/4)]    [∵ cos (-θ) = cos θ and sin (-θ) =- sin θ]  

Hence, option (2) is correct.


Discussion

No Comment Found

Related InterviewSolutions