صفحة 212 - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

{ "language": "ar", "direction": "rtl", "page_context": { "page_title": "التمثيل البياني للدوال المثلثية الأساسية", "page_type": "lesson_content", "main_topics": [ "الدوال المثلثية", "التمثيل البياني للدوال" ], "headers": [ "التمثيل البياني للدوال المثلثية الأساسية", "بعض قيم الدوال المثلثية للزوايا الخاصة", "دوال في دائرة الوحدة" ], "has_questions": false, "has_formulas": true, "has_examples": true, "has_visual_elements": true }, "sections": [ { "order": 1, "type": "header", "title": "التمثيل البياني للدوال المثلثية الأساسية", "content": "التمثيل البياني للدوال المثلثية الأساسية", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 2, "type": "main_content", "content": "الدالة\nالتمثيل البياني\n$y = \\tan \\theta$\n$y = \\cos \\theta$\n$y = \\sin \\theta$", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 3, "type": "header", "title": "بعض قيم الدوال المثلثية للزوايا الخاصة", "content": "بعض قيم الدوال المثلثية للزوايا الخاصة", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 4, "type": "main_content", "content": "$30^{\\circ}, 60^{\\circ}, 90^{\\circ}$\n$\sin 30^{\\circ} = \\frac{1}{2}$\n$\sin 60^{\\circ} = \\frac{\\sqrt{3}}{2}$\n$\cos 30^{\\circ} = \\frac{\\sqrt{3}}{2}$\n$\cos 60^{\\circ} = \\frac{1}{2}$\n$\tan 30^{\\circ} = \\frac{\\sqrt{3}}{3}$\n$\tan 60^{\\circ} = \\sqrt{3}$\n$45^{\\circ}, 45^{\\circ}, 90^{\\circ}$\n$\sin 45^{\\circ} = \\frac{\\sqrt{2}}{2}$\n$\cos 45^{\\circ} = \\frac{\\sqrt{2}}{2}$\n$\tan 45^{\\circ} = 1$", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 5, "type": "header", "title": "دوال في دائرة الوحدة", "content": "دوال في دائرة الوحدة", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 6, "type": "main_content", "content": "إذا قطع ضلع الانتهاء للزاوية $\\theta$ المرسومة في الوضع القياسي دائرة الوحدة في النقطة $(x, y)$ فإن $y = \\sin \\theta$ و $x = \\cos \\theta$. أي أن: $P(x, y) = P(\\cos \\theta, \\sin \\theta)$.\nمثال: إذا كانت $\\theta = 120^{\\circ}$ فإن $P(x, y) = P(\\cos 120^{\\circ}, \\sin 120^{\\circ})$", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 7, "type": "footer", "content": "وزارة التعليم\nMinistry of Education\n2025 - 1447\nالصيغ والرموز\n212", "content_classification": "METADATA" } ], "visual_elements": [ { "index": 0, "label": "y = tan θ", "question_number": null, "type": "graph", "location": "top left of page", "coordinate_system": "Standard Cartesian grid, units marked at -90, 0, 90, 180, 270, 360, 450 degrees on x-axis and -2, -1, 0, 1, 2 on y-axis. Grid lines are spaced at 90 degrees intervals on x-axis and 1 unit on y-axis.", "shape": "periodic, discontinuous wave-like curves with vertical asymptotes", "function": "y = tan θ", "description": "Three cycles of the tangent function are shown. Each cycle has vertical asymptotes at odd multiples of 90 degrees (e.g., -90, 90, 270, 450 degrees). The function increases from negative infinity to positive infinity within each interval between asymptotes.", "axes_labels": { "x_axis": "θ (degrees)", "y_axis": "y" }, "axes_ranges": { "x_min": -90, "x_max": 450, "y_min": -2, "y_max": 2 }, "endpoints": [], "critical_points": [], "y_intercept": { "x": 0, "y": 0, "description": "crosses y-axis at the origin (0, 0)" }, "end_behavior": { "left": "arrow pointing down-left: x→-90°, y→-∞ (approaching asymptote)", "right": "arrow pointing up-right: x→90°, y→+∞ (approaching asymptote)" }, "key_points": [ "Asymptotes at θ = -90°, 90°, 270°, 450°", "Passes through (0, 0), (180, 0), (360, 0)" ], "educational_context": "Illustrates the tangent function's behavior, periodicity, and asymptotes." }, { "index": 1, "label": "y = cos θ", "question_number": null, "type": "graph", "location": "top center of page", "coordinate_system": "Standard Cartesian grid. Units marked at 0, 90, 180, 270, 360, 450, 540 degrees on x-axis and -1, 0, 1 on y-axis. Grid lines are spaced at 90 degree intervals on x-axis and 1 unit on y-axis.", "shape": "continuous, periodic wave-like curve", "function": "y = cos θ", "description": "One and a half cycles of the cosine function are shown, starting from its maximum value at θ=0.", "axes_labels": { "x_axis": "θ (degrees)", "y_axis": "y" }, "axes_ranges": { "x_min": 0, "x_max": 540, "y_min": -1, "y_max": 1 }, "endpoints": [], "critical_points": [ { "type": "local_max", "coordinates": { "x": 0, "y": 1 }, "description": "maximum value at (0, 1)" }, { "type": "local_min", "coordinates": { "x": 180, "y": -1 }, "description": "minimum value at (180, -1)" }, { "type": "local_max", "coordinates": { "x": 360, "y": 1 }, "description": "maximum value at (360, 1)" }, { "type": "local_min", "coordinates": { "x": 540, "y": -1 }, "description": "minimum value at (540, -1)" } ], "y_intercept": { "x": 0, "y": 1, "description": "crosses y-axis at (0, 1)" }, "end_behavior": { "left": "N/A (starts at x=0)", "right": "arrow pointing up-right: x→+∞, y→+∞ (implied continuation of pattern)" }, "key_points": [ "Starts at (0, 1)", "Crosses x-axis at (90, 0), (270, 0), (450, 0)", "Reaches minimum at (180, -1), (540, -1)", "Reaches maximum at (0, 1), (360, 1)" ], "educational_context": "Illustrates the cosine function's behavior, periodicity, amplitude, and phase shift." }, { "index": 2, "label": "y = sin θ", "question_number": null, "type": "graph", "location": "top right of page", "coordinate_system": "Standard Cartesian grid. Units marked at 0, 90, 180, 270, 360, 450, 540 degrees on x-axis and -1, 0, 1 on y-axis. Grid lines are spaced at 90 degree intervals on x-axis and 1 unit on y-axis.", "shape": "continuous, periodic wave-like curve", "function": "y = sin θ", "description": "One and a half cycles of the sine function are shown, starting from its value at θ=0.", "axes_labels": { "x_axis": "θ (degrees)", "y_axis": "y" }, "axes_ranges": { "x_min": 0, "x_max": 540, "y_min": -1, "y_max": 1 }, "endpoints": [], "critical_points": [ { "type": "local_max", "coordinates": { "x": 90, "y": 1 }, "description": "maximum value at (90, 1)" }, { "type": "local_min", "coordinates": { "x": 270, "y": -1 }, "description": "minimum value at (270, -1)" }, { "type": "local_max", "coordinates": { "x": 450, "y": 1 }, "description": "maximum value at (450, 1)" } ], "y_intercept": { "x": 0, "y": 0, "description": "crosses y-axis at the origin (0, 0)" }, "end_behavior": { "left": "N/A (starts at x=0)", "right": "arrow pointing up-right: x→+∞, y→+∞ (implied continuation of pattern)" }, "key_points": [ "Starts at (0, 0)", "Crosses x-axis at (0, 0), (180, 0), (360, 0), (540, 0)", "Reaches maximum at (90, 1), (450, 1)", "Reaches minimum at (270, -1)" ], "educational_context": "Illustrates the sine function's behavior, periodicity, amplitude, and phase shift." }, { "index": 3, "label": "Right Triangle with 30-60-90 angles", "question_number": null, "type": "diagram", "location": "middle right of page", "coordinate_system": "Not applicable (geometric diagram)", "shape": "right-angled triangle", "function": null, "description": "A right-angled triangle with angles 30°, 60°, and 90°. The side opposite the 30° angle is labeled 'x'. The side opposite the 60° angle is labeled 'x√3'. The hypotenuse (opposite the 90° angle) is labeled '2x'.", "axes_labels": null, "axes_ranges": null, "endpoints": [], "critical_points": [], "y_intercept": null, "end_behavior": null, "key_points": [ "Sides are in ratio x : x√3 : 2x for angles 30° : 60° : 90° respectively." ], "educational_context": "Shows the side length ratios for a 30-60-90 special right triangle, useful for trigonometric calculations." }, { "index": 4, "label": "Right Triangle with 45-45-90 angles", "question_number": null, "type": "diagram", "location": "bottom right of page", "coordinate_system": "Not applicable (geometric diagram)", "shape": "right-angled triangle", "function": null, "description": "A right-angled triangle with angles 45°, 45°, and 90°. The two legs (opposite the 45° angles) are labeled 'x'. The hypotenuse (opposite the 90° angle) is labeled 'x√2'.", "axes_labels": null, "axes_ranges": null, "endpoints": [], "critical_points": [], "y_intercept": null, "end_behavior": null, "key_points": [ "Sides are in ratio x : x : x√2 for angles 45° : 45° : 90° respectively.", "It is an isosceles right triangle." ], "educational_context": "Shows the side length ratios for a 45-45-90 special right triangle, useful for trigonometric calculations." }, { "index": 5, "label": "Unit Circle Diagram", "question_number": null, "type": "diagram", "location": "middle left of page", "coordinate_system": "Standard Cartesian grid with origin O. Unit circle centered at origin with radius 1.", "shape": "circle", "function": null, "description": "A unit circle centered at the origin (0,0) with points marked on the axes: (1,0) on the positive x-axis, (0,1) on the positive y-axis, (-1,0) on the negative x-axis, and (0,-1) on the negative y-axis. An angle θ is shown in standard position, with its terminal side intersecting the circle at point P(cos θ, sin θ).", "axes_labels": { "x_axis": "x", "y_axis": "y" }, "axes_ranges": { "x_min": -1.5, "x_max": 1.5, "y_min": -1.5, "y_max": 1.5 }, "endpoints": [], "critical_points": [], "y_intercept": null, "end_behavior": null, "key_points": [ "The point P on the unit circle has coordinates (cos θ, sin θ).", "The radius of the unit circle is 1." ], "educational_context": "Defines sine and cosine in the context of the unit circle, relating angles to coordinates on the circle." } ] }