Question

Represent the following number in the polar form:

$(1 + i \sqrt{3})$

Accepted Answer

Given, $(1 + i \sqrt{3})$ $\to (i)$

Let the polar form be

$z = r (\cos θ + i \sin θ)$

$\Rightarrow z = r \cos θ + i r \sin θ \to (i i)$

Comparing equations $(i)$ and $(i i)$ , we have

$1 + i \sqrt{3} = r \cos θ + i r \sin θ$

Two complex numbers are equal if and only if their real and imaginary parts are respectively equal to each other.

$\Rightarrow 1 = r \cos θ \to (i i i)$ and $\sqrt{3} = r \sin θ \to (i v)$

Now, squaring and adding equations $(i i i)$ and $(i v)$ , we have

$3 + 1 = r^{2} (\cos^{2} θ + \sin^{2} θ)$

$\Rightarrow r^{2} = 4$

$\Rightarrow r = 2 [∵ r > 0]$

Now, $\sin θ = \frac{\sqrt{3}}{2}$ and $\cos θ = \frac{1}{2} [using (i i i) and (i v)]$

Also, $\sin θ > 0$ and $\cos θ > 0$

$\Rightarrow θ$ lies in the first quadrant

$∴ \tan α = |\tan θ| = \sqrt{3}$

$\Rightarrow α = \frac{π}{3}$

And, in the first quadrant $θ = α$

$\Rightarrow θ = (\frac{π}{3})$

Hence, the required polar form of the complex number is

$z = 2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))$ $[Using (i i)]$

R S Aggarwal Solutions for Exercise 3: EXERCISE 2C

R S Aggarwal Mathematics Solutions for Exercise - R S Aggarwal Solutions for Exercise 3: EXERCISE 2C

Questions from R S Aggarwal Solutions for Exercise 3: EXERCISE 2C with Hints & Solutions

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