Find the modulus and the argument of the complex number z = – 1 – $$i\sqrt{3}$$

Asked by Pragya Singh | 1 year ago |  85

##### Solution :-

The complex number is

$$z= – 1 – i\sqrt{3}$$

Let rcosθ = -1 and rsinθ = $$-\sqrt{3}$$

(rcosθ)2 + (rsinθ)2$$(-1)^2+(- \sqrt{3})^2$$

$$r^2(cos^2θ) + (sin^2θ)=1+3$$

r2 = 4  $$[ cos^2θ + sin^2θ=1]$$

$$r=\sqrt{4}=2$$  [ Conventionally, r > 0]

Modulus = 2

2cosθ = -1 and 2sinθ = $$-\sqrt{3}$$

$$cosθ =\dfrac{-1}{2}and \;sinθ=\dfrac{-\sqrt{3}}{2}$$

Since both the values of sinθ and cosθ negative and sinθ and cosθ are negative in 3rd quadrant,

Argument = $$-(\pi-\dfrac{\pi}{3})=\dfrac{-2\pi}{3}$$

Thus, the modulus and argument of the complex number $$– 1 – i\sqrt{3}$$  are 2 and $$\dfrac{-2\pi}{3}$$ Respectively

Answered by Abhisek | 1 year ago

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