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Home » The general formula for the \( n \)-th term of an arithmetic sequence is: - ToelettAPP

The general formula for the \( n \)-th term of an arithmetic sequence is: - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 السؤال: من بين جميع جذور المعادلة $z^4 - z^2 + 1 = 0$, يمكن التعبير عن الجزء التخيلي الأكبر من جذر على شكل $\sin \theta$, حيث $\theta \in [0^\circ, 90^\circ]$. أوجد $\theta$.

📌 z^4 - z^2 + 1 = 0

📌 لنفترض $w = z^2$, إذن:

📌 w^2 - w + 1 = 0 \implies w = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm i\sqrt{3}}{2}

📌 إذن، $z^2 = \frac{1 \pm i\sqrt{3}}{2}$. الجذور التربيعية لـ $\frac{1 + i\sqrt{3}}{2}$ (التي تُعتبر $e^{i\pi/3}$) هي:

📌 z = e^{i\pi/6} \quad \text{و} \quad z = e^{i5\pi/6}

📌 الأجزاء التخيلية هي $\sin \frac{\pi}{6} = \frac{1}{2}$ و $\sin \frac{5\pi}{6} = \frac{1}{2}$. للجذور من $z^2 = \frac{1 - i\sqrt{3}}{2} = e^{-i\pi/3}$, فإن الجذور التربيعية هي:

📌 z = e^{-i\pi/6} \quad \text{و} \quad z = e^{-i5\pi/6}

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