QuestionMay 5, 2026

Convert into a complex number: 2(cos(4pi )/(3)+isin(4pi )/(3)) -sqrt (3)-i sqrt (3)-i -1-isqrt (3) 1+isqrt (3)

Convert into a complex number: 2(cos(4pi )/(3)+isin(4pi )/(3)) -sqrt (3)-i sqrt (3)-i -1-isqrt (3) 1+isqrt (3)
Convert into a complex number: 2(cos(4pi )/(3)+isin(4pi )/(3)) -sqrt (3)-i sqrt (3)-i -1-isqrt (3) 1+isqrt (3)

Solution
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Answer

-1 - i\sqrt{3} Explanation 1. Apply Euler's formula Use **re^{i\theta} = r(\cos\theta + i\sin\theta)**. Here r=2, \theta = \frac{4\pi}{3}. So: 2\left(\cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3}\right). 2. Evaluate trigonometric values \cos\frac{4\pi}{3} = -\frac{1}{2}, \sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}. 3. Multiply by r 2\left(-\frac{1}{2} + i\left(-\frac{\sqrt{3}}{2}\right)\right) = -1 - i\sqrt{3}.

Explanation

1. Apply Euler's formula <br /> Use **$re^{i\theta} = r(\cos\theta + i\sin\theta)$**. Here $r=2$, $\theta = \frac{4\pi}{3}$. <br />So: $2\left(\cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3}\right)$.<br /><br />2. Evaluate trigonometric values <br /> $\cos\frac{4\pi}{3} = -\frac{1}{2}$, $\sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$.<br /><br />3. Multiply by r <br /> $2\left(-\frac{1}{2} + i\left(-\frac{\sqrt{3}}{2}\right)\right) = -1 - i\sqrt{3}$.
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