if-z-z-1-3-i-find-

Question Number 149142 by mathdanisur last updated on 03/Aug/21

i f z + ∣ z ∣ = 1 + \sqrt{3} i

f i n d φ = ?

Answered by liberty last updated on 03/Aug/21

z = a + b i

∣ z ∣= \sqrt{a^{2} + b^{2}}

\Rightarrow a + b i + \sqrt{a^{2} + b^{2}} = 1 + \sqrt{3} i

{\begin{cases} b = \sqrt{3} \\ a + \sqrt{a^{2} + 3} = 1 \Rightarrow \sqrt{a^{2} + 3} = 1 - a \end{cases}

\Rightarrow a^{2} + 3 = a^{2} - 2 a + 1

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

\tan φ = \frac{\sqrt{3}}{- 1} = - \sqrt{3}

φ = - \frac{π}{3}

Commented by mathdanisur last updated on 03/Aug/21

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