Find the principal argument of 1- i -3-

Given that, (1+i√(3)) ⇒ z=1 3+2i√(3) ⇒ z= 2+2i√(3) ⇒ tan α=|2√(3)/ 2|=| √(3)|=√(3) [∵ tan α=|Im(z)/Re(z)| ] ⇒ tan α=tan π/3⇒ α=π/3 ∵ Re (z) lt;0 a ...

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Standard XII

Mathematics

Properties of Argument

Find the prin...

Question

Find the principal argument of $(1 + i \sqrt{3})$ .

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Solution

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

$\Rightarrow z = 1 - 3 + 2 i \sqrt{3} \Rightarrow z = - 2 + 2 i \sqrt{3} \Rightarrow t a n α = ∣ ∣ ∣ \frac{2 \sqrt{3}}{- 2} ∣ ∣ ∣ = | - \sqrt{3} | = \sqrt{3} [∵ t a n α = ∣ ∣ \frac{I m (z)}{R e (z)} ∣ ∣] \Rightarrow t a n α = t a n \frac{π}{3} \Rightarrow α = \frac{π}{3} ∵ R e (z) < 0 and I m (z) > 0 \Rightarrow a r g (z) = π - \frac{π}{3} \Rightarrow = \frac{2 π}{3}$

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Find the principal argument of (1+i√3).

Given that, (1+i√3) ⇒ z=1−3+2i√3⇒ z=−2+2i√3⇒ tan α=∣∣∣2√3−2∣∣∣=|−√3|=√3 [∵ tan α=∣∣Im(z)Re(z)∣∣]⇒ tan α=tan π3⇒ α=π3∵ Re (z)<0 and Im (z)>0⇒ arg(z)=π−π3⇒ =2π3

Find the principal argument of $(1 + i \sqrt{3})$ .