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Home » \[ \fracd80 + \fracd60 = 7 \] - Dachbleche24

\[ \fracd80 + \fracd60 = 7 \] - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 = 1 \cdot 105 + 60

📌 = 1 \cdot 60 + 45

📌 = 1 \cdot 45 + 15

📌 = 3 \cdot 15 + 0

📌 عندما يصبح الباقي صفرًا، فإن القاسم المشترك الأكبر هو آخر باقي غير صفري، وهو $15$.

📌 إذن، $\gcd(105, 165) = \boxed{15}$.السؤال: أوجد مجموع جميع الزوايا $z \in [0^\circ, 360^\circ]$ التي تحقق $\sin(2z) = \frac{\sqrt{3}}{2}$.

📌 الحل: لحل المعادلة $\sin(2z) = \frac{\sqrt{3}}{2}$، نحدد أولاً الزوايا التي تساوي فيها دالة الجيب $\frac{\sqrt{3}}{2}$. هذه الزوايا في الربعين الأول والثاني ضمن الفترة $[0^\circ, 360^\circ]$ هي $60^\circ$ و $120^\circ$. بما أن وسيط دالة الجيب ضمن المعادلة هو $2z$، فإننا نحل لإيجاد $2z$:

📌 2z = 60^\circ \quad \text{أو} \quad 2z = 120^\circ

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