∵ z= 1+√( 3) ⇒ z= 1+i√(3) Let θ nbsp;be acute angle such that tan⁡ θ=|Im(z)||Re(z)| ⇒tanθ=|√(3)|| 1| ⇒tanθ=√(3) ⇒θ=π3 ∵ z nbsp;lies on the 2nd quadra ...

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Standard XI

Mathematics

Sign of Trigonometric Ratios in Different Quadrants

If z=-1+√-3 t...

Question

If $z = - 1 + \sqrt{- 3}$ then $a r g (z) =$ ___________

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Solution

$∵ z = - 1 + \sqrt{- 3}$

$\Rightarrow z = - 1 + i \sqrt{3}$

Let $θ$ be acute angle such that-

$tan θ = \frac{| I m (z) |}{| R e (z) |}$

$\Rightarrow tan θ = \frac{| \sqrt{3} |}{| - 1 |}$

$\Rightarrow tan θ = \sqrt{3}$
$\Rightarrow θ = \frac{π}{3}$
$∵ z$ lies on the $2 n d$ quadrant\)

$\Rightarrow a r g (z) = π - \frac{π}{3}$

$\Rightarrow a r g (z) = \frac{2 π}{3}$

$∴$ If $z = - 1 + \sqrt{- 3}$ , then $a r g (z) = \frac{2 π}{3}$

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If z=−1+√−3 then arg(z)= ___________

∵z=−1+√−3 ⇒z=−1+i√3 Let θ be acute angle such that- tanθ=|Im(z)||Re(z)| ⇒tanθ=|√3||−1| ⇒tanθ=√3 ⇒θ=π3 ∵z lies on the 2nd quadrant\) ⇒arg(z)=π−π3 ⇒arg(z)=2π3 ∴ If z=−1+√−3 , then arg(z)=2π3

If $z = - 1 + \sqrt{- 3}$ then $a r g (z) =$ ___________