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# complex number

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given that $$\left | z - 2 \right |=2$$ and $$arg(z-2) = -\pi/3$$

Express z in the form a+bi where a and b are real numbers

Feb 20, 2019

$$\text{consider }\tilde{z} = z-2\\ |\tilde{z}|=2 \text{ and }\arg(\tilde{z}) = -\dfrac{\pi}{3}\\ \tilde{z} = 2 \cos\left(-\dfrac{\pi}{3}\right) + i 2 \sin\left(-\dfrac{\pi}{3}\right)\\ \tilde{z} = 1-i \sqrt{3},~\text{ and thus}\\ z = \tilde{z}+2 = 3 - i \sqrt{3}$$