مراجعة تراكمية ومسائل مهارات التفكير العليا - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

{ "language": "ar", "direction": "rtl", "page_context": { "page_title": "مراجعة تراكمية ومسائل مهارات التفكير العليا", "page_type": "exercises", "main_topics": [ "مراجعة تراكمية", "مسائل مهارات التفكير العليا", "المتطابقات والمعادلات المثلثية" ], "headers": [ "مراجعة تراكمية", "مسائل مهارات التفكير العليا", "تدريب على اختبار", "الفصل 3 المتطابقات والمعادلات المثلثية" ], "has_questions": true, "has_formulas": true, "has_examples": false, "has_visual_elements": true }, "sections": [ { "order": 1, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "مسائل مهارات التفكير العليا", "content": "مسائل مهارات التفكير العليا", "associated_visual_elements": [] }, { "order": 2, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "27", "content": "اكتشف الخطأ: يحاول سعيد وسلمان حساب القيمة الدقيقة لـ: sin 15°. هل إجابة أي منهما صحيحة؟ برر إجابتك.\nسعيد\nsin (A - B) = sin A cos B - cos A sin B\nsin (45 - 30) = sin 45 cos 30 - cos 45 sin 30\n= (√2/2) * (√3/2) - (√2/2) * (1/2)\n= √6/4 - √2/4\n= (√6 - √2) / 4\nسلمان\nsin (A/2) = √(1 - cos A) / 2\nsin (30/2) = √(1 - cos 30) / 2\nsin 15° = √(1 - (√3/2)) / 2\n= √( (2 - √3) / 2 ) / 2\n= √(2 - √3) / 4\n= 0.5", "associated_visual_elements": [] }, { "order": 3, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": true }, "title": "28", "content": "تحد: استعمل دائرة الوحدة أدناه، والشكل المرسوم داخلها؛ لتبرهن أن: tan (θ/2) = sin θ / (1 + cos θ)", "associated_visual_elements": [ 0 ] }, { "order": 4, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": true }, "title": "29", "content": "اكتب: اكتب فقرة مختصرة تبين الشروط اللازم توافرها كي تستعمل كلا من المتطابقات الثلاث لـ cos 2θ.", "associated_visual_elements": [] }, { "order": 5, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": true }, "title": "30", "content": "برهان: استعمل الصيغة (A + B) sin لاشتقاق صيغة sin 2θ، واستعمل الصيغة (A + B) cos لاشتقاق صيغة cos 2θ.", "associated_visual_elements": [] }, { "order": 6, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": true }, "title": "31", "content": "تعبير: اشتق المتطابقات المثلثية لنصف الزاوية من المتطابقات المثلثية لضعف الزاوية.", "associated_visual_elements": [] }, { "order": 7, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": true }, "title": "32", "content": "مسألة مفتوحة: ضرب لاعب جولف كرة عدة مرات بسرعة ابتدائية مقدارها ft/s 115، ولنفترض أن المسافة d التي قطعتها الكرة في كل مرة تُعطى بالصيغة d = (2v² sin θ cos θ) / g. فسر لماذا تكون المسافة العظمى عندما °45 = θ. (g = 32 ft/s²)", "associated_visual_elements": [] }, { "order": 8, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "مراجعة تراكمية", "content": "مراجعة تراكمية", "associated_visual_elements": [] }, { "order": 9, "type": "instruction", "content_classification": "FIGURE_REFERENCE", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": true }, "title": null, "content": "أثبت صحة كل من المتطابقات الآتية: (الدرس 3-2)", "associated_visual_elements": [] }, { "order": 10, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "33", "content": "cot θ + sec θ = (cos² θ + sin θ) / (sin θ cos θ)", "associated_visual_elements": [] }, { "order": 11, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "34", "content": "sin² θ + tan² θ = (1 - cos² θ) + (sec² θ / csc² θ)", "associated_visual_elements": [] }, { "order": 12, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "35", "content": "(sin θ - cos θ)² = 1 - 2 sin θ cos θ", "associated_visual_elements": [] }, { "order": 13, "type": "instruction", "content_classification": "FIGURE_REFERENCE", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": true }, "title": null, "content": "أوجد القيمة الدقيقة لكل مما يأتي: (الدرس 3-3)", "associated_visual_elements": [] }, { "order": 14, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "36", "content": "sin 135°", "associated_visual_elements": [] }, { "order": 15, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "37", "content": "cos 105°", "associated_visual_elements": [] }, { "order": 16, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "38", "content": "sin 285°", "associated_visual_elements": [] }, { "order": 17, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "39", "content": "cos 210°", "associated_visual_elements": [] }, { "order": 18, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "40", "content": "sin (-240°)", "associated_visual_elements": [] }, { "order": 19, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "41", "content": "cos (-120°)", "associated_visual_elements": [] }, { "order": 20, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": false, "has_instruction_words": false }, "title": "42", "content": "cos 78° cos 18° + sin 78° sin 18°", "associated_visual_elements": [] }, { "order": 21, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "تدريب على اختبار", "content": "تدريب على اختبار", "associated_visual_elements": [] }, { "order": 22, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": true, "has_instruction_words": false }, "title": "43", "content": "أوجد القيمة الدقيقة لـ tan (θ/2) إذا كان cos θ = √3/2 ; 0 < θ < 90°.", "associated_visual_elements": [], "sub_questions": [ { "number": "A", "question": "√(7 - 4√3)", "visual_element_index": null }, { "number": "B", "question": "√3 - 2", "visual_element_index": null }, { "number": "C", "question": "√3 / 3", "visual_element_index": null }, { "number": "D", "question": "√3", "visual_element_index": null } ] }, { "order": 23, "type": "exercise", "content_classification": "QUESTION", "question_indicators": { "has_question_words": true, "has_numbering": true, "has_multiple_choice": true, "has_instruction_words": false }, "title": "44", "content": "معادلة الدالة الممثلة بيانيًّا في الشكل أدناه هي:", "associated_visual_elements": [ 1 ], "sub_questions": [ { "number": "A", "question": "y = 3 cos 2θ", "visual_element_index": 1 }, { "number": "B", "question": "y = (1/3) cos 2θ", "visual_element_index": 1 }, { "number": "C", "question": "y = 3 cos (1/2)θ", "visual_element_index": 1 }, { "number": "D", "question": "y = (1/3) cos (1/2)θ", "visual_element_index": 1 } ] }, { "order": 24, "type": "metadata", "content_classification": "METADATA", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": null, "content": "وزارة التعليم\nMinistry of Education\n2025 - 1447", "associated_visual_elements": [] }, { "order": 25, "type": "footer", "content_classification": "METADATA", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": null, "content": "الفصل 3 المتطابقات والمعادلات المثلثية 156", "associated_visual_elements": [] } ], "visual_elements": [ { "index": 0, "label": "دائرة الوحدة", "question_number": "28", "type": "diagram", "location": "right side of page", "coordinate_system": "N/A (geometric diagram)", "shape": "circle with inscribed right triangle", "title": null, "function": null, "description": "A unit circle centered at the origin O. A point P is on the circle in the first quadrant. A perpendicular line segment PA is drawn from P to the x-axis, forming a right-angled triangle OAP. The angle between the positive x-axis and the line segment OP is labeled θ. Another point B is on the circle, and the line segment OB forms an angle with the positive x-axis that is half of θ (θ/2). Point D is on the negative y-axis. The diagram illustrates geometric relationships within a unit circle to prove a trigonometric identity.", "axes_labels": { "x_axis": null, "y_axis": null }, "axes_ranges": { "x_min": null, "x_max": null, "y_min": null, "y_max": null }, "endpoints": [], "critical_points": [], "y_intercept": null, "end_behavior": { "left": "N/A", "right": "N/A" }, "key_points": [ { "x": 0, "y": 0, "label": "O (Origin)" }, { "x": "x_p", "y": 0, "label": "A (on x-axis)" }, { "x": "x_p", "y": "y_p", "label": "P (on circle)" }, { "x": "x_b", "y": "y_b", "label": "B (on circle)" }, { "x": 0, "y": "y_d", "label": "D (on negative y-axis)" } ], "data_description": "Geometric representation of angles and coordinates on a unit circle.", "key_values": [ "Angle θ", "Angle θ/2" ], "numeric_data": null, "table_structure": null, "educational_context": "Used to visually demonstrate the half-angle identity for tangent.", "estimated": false }, { "index": 1, "label": "Graph for Question 44", "question_number": "44", "type": "graph", "location": "left side of page", "coordinate_system": "Standard Cartesian grid. On the x-axis, 4 grid squares represent π units, so 1 square = π/4 units. On the y-axis, 2 grid squares represent 1 unit, so 1 square = 0.5 units.", "shape": "continuous wave-like curve, resembling a cosine function", "title": null, "function": "y = cos(2θ)", "description": "A periodic trigonometric function oscillating between y = -1 and y = 1. The curve starts at a maximum at (0,1), decreases to a minimum at (π/2, -1), and returns to a maximum at (π, 1), completing one full cycle. The period is π and the amplitude is 1.", "axes_labels": { "x_axis": "θ", "y_axis": "y" }, "axes_ranges": { "x_min": -2.5 * 3.141592653589793, "x_max": 2.5 * 3.141592653589793, "y_min": -1.5, "y_max": 1.5 }, "endpoints": [], "critical_points": [ { "type": "local_max", "coordinates": { "x": 0, "y": 1 }, "description": "Highest peak at the y-axis. From origin (0,0): 0 squares left/right, 2 squares up (2 * 0.5 = 1)." }, { "type": "local_min", "coordinates": { "x": "π/2", "y": -1 }, "description": "Lowest trough. From origin (0,0): 2 squares right (2 * π/4 = π/2), 2 squares down (2 * 0.5 = -1)." }, { "type": "local_max", "coordinates": { "x": "π", "y": 1 }, "description": "Next peak. From origin (0,0): 4 squares right (4 * π/4 = π), 2 squares up (2 * 0.5 = 1)." }, { "type": "local_min", "coordinates": { "x": "3π/2", "y": -1 }, "description": "Next trough. From origin (0,0): 6 squares right (6 * π/4 = 3π/2), 2 squares down (2 * 0.5 = -1)." }, { "type": "local_max", "coordinates": { "x": "2π", "y": 1 }, "description": "Next peak. From origin (0,0): 8 squares right (8 * π/4 = 2π), 2 squares up (2 * 0.5 = 1)." }, { "type": "local_min", "coordinates": { "x": "-π/2", "y": -1 }, "description": "Trough to the left of origin. From origin (0,0): 2 squares left (2 * π/4 = -π/2), 2 squares down (2 * 0.5 = -1)." }, { "type": "local_max", "coordinates": { "x": "-π", "y": 1 }, "description": "Peak to the left of origin. From origin (0,0): 4 squares left (4 * π/4 = -π), 2 squares up (2 * 0.5 = 1)." }, { "type": "local_min", "coordinates": { "x": "-3π/2", "y": -1 }, "description": "Next trough to the left. From origin (0,0): 6 squares left (6 * π/4 = -3π/2), 2 squares down (2 * 0.5 = -1)." }, { "type": "local_max", "coordinates": { "x": "-2π", "y": 1 }, "description": "Next peak to the left. From origin (0,0): 8 squares left (8 * π/4 = -2π), 2 squares up (2 * 0.5 = 1)." } ], "y_intercept": { "x": 0, "y": 1, "description": "The curve crosses the y-axis at its maximum point (0, 1)." }, "end_behavior": { "left": "arrow pointing up-left, indicating x→-∞, y oscillates between -1 and 1", "right": "arrow pointing up-right, indicating x→+∞, y oscillates between -1 and 1" }, "key_points": [], "data_description": "The graph displays a periodic oscillation characteristic of a cosine function with an amplitude of 1 and a period of π.", "key_values": [ "Amplitude: 1", "Period: π" ], "numeric_data": null, "table_structure": null, "educational_context": "This graph represents a trigonometric function, y = cos(2θ), and is used to identify the correct equation from multiple-choice options. The domain is (-∞, +∞) and the range is [-1, 1].", "estimated": false } ] }