الصيغ - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

--- SECTION: الصيغ --- الصيغ --- SECTION: متطابقات الدوال الزوجية أو الفردية --- sin (-θ) = -sin θ cos (-θ) = cos θ tan (-θ) = -tan θ csc (-θ) = -csc θ sec (-θ) = sec θ cot (-θ) = -cot θ --- SECTION: متطابقات المجموع والفرق --- cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B tan (A + B) = (tan A + tan B) / (1 - tan A tan B) tan (A - B) = (tan A - tan B) / (1 + tan A tan B) --- SECTION: متطابقات ضعف الزاوية --- cos 2θ = cos² θ - sin² θ cos 2θ = 2 cos² θ - 1 cos 2θ = 1 - 2 sin² θ sin 2θ = 2 sin θ cos θ tan 2θ = (2 tan θ) / (1 - tan² θ) --- SECTION: متطابقات نصف الزاوية --- sin (θ/2) = ±√( (1 - cos θ) / 2 ) cos (θ/2) = ±√( (1 + cos θ) / 2 ) tan (θ/2) = ±√( (1 - cos θ) / (1 + cos θ) ) --- SECTION: التمثيل البياني للدوال الرئيسة (الأم) --- التمثيل البياني للدوال الرئيسة (الأم) --- SECTION: قيم الدوال المثلثية لبعض الزوايا الخاصة --- قيم الدوال المثلثية لبعض الزوايا الخاصة --- SECTION: Footer Information --- وزارة التعليم الصيغ والرموز 211 of 211 2025 - 1447 --- VISUAL CONTEXT --- **TABLE**: متطابقات الدوال المثلثية Description: A table listing various trigonometric identities categorized by type: Even/Odd Functions, Sum/Difference, Double Angle, and Half Angle identities. Table Structure: Headers: متطابقات الدوال الزوجية أو الفردية | متطابقات المجموع والفرق | متطابقات ضعف الزاوية | متطابقات نصف الزاوية Rows: Row 1: sin (-θ) = -sin θ | cos (A + B) = cos A cos B - sin A sin B | cos 2θ = cos² θ - sin² θ | sin (θ/2) = ±√( (1 - cos θ) / 2 ) Row 2: cos (-θ) = cos θ | cos (A - B) = cos A cos B + sin A sin B | cos 2θ = 2 cos² θ - 1 | cos (θ/2) = ±√( (1 + cos θ) / 2 ) Row 3: tan (-θ) = -tan θ | sin (A + B) = sin A cos B + cos A sin B | cos 2θ = 1 - 2 sin² θ | tan (θ/2) = ±√( (1 - cos θ) / (1 + cos θ) ) Row 4: csc (-θ) = -csc θ | sin (A - B) = sin A cos B - cos A sin B | sin 2θ = 2 sin θ cos θ | EMPTY Row 5: sec (-θ) = sec θ | tan (A + B) = (tan A + tan B) / (1 - tan A tan B) | tan 2θ = (2 tan θ) / (1 - tan² θ) | EMPTY Row 6: cot (-θ) = -cot θ | tan (A - B) = (tan A - tan B) / (1 + tan A tan B) | EMPTY | EMPTY Empty cells: Cells are empty where a category has fewer formulas than the maximum number of formulas in any other category. Calculation needed: No calculations needed, this is a reference table of identities. Context: Provides a comprehensive list of fundamental trigonometric identities for reference in mathematics. **GRAPH**: الدالة الثابتة Description: A Cartesian coordinate graph showing a horizontal line, representing a constant function f(x)=c. X-axis: x Y-axis: y Data: A horizontal line parallel to the x-axis, indicating that the y-value (output) is constant for all x-values (input). The line is labeled f(x)=c. Context: Illustrates the graphical representation of a constant function, f(x) = c. **GRAPH**: الدالة التربيعية Description: A Cartesian coordinate graph showing a parabola opening upwards, representing a quadratic function y=x². X-axis: x Y-axis: y Data: A U-shaped curve (parabola) with its vertex at the origin (O) and opening upwards, symmetric about the y-axis. The function is labeled y=x². Context: Illustrates the graphical representation of a basic quadratic function, y = x². **GRAPH**: دالة القيمة المطلقة Description: A Cartesian coordinate graph showing a V-shaped curve, representing an absolute value function y=|x|. X-axis: x Y-axis: y Data: A V-shaped graph with its vertex at the origin (O) and opening upwards, symmetric about the y-axis. The function is labeled y=|x|. Context: Illustrates the graphical representation of an absolute value function, y = |x|. **GRAPH**: الدالة المحايدة Description: A Cartesian coordinate graph showing a straight line passing through the origin, representing the identity function y=x. X-axis: x Y-axis: y Data: A straight line passing through the origin (O) with a positive slope (slope of 1), extending infinitely in both directions. The function is labeled y=x. Context: Illustrates the graphical representation of the identity function, y = x. **GRAPH**: الدالة التكعيبية Description: A Cartesian coordinate graph showing a curve representing a cubic function f(x)=x³. X-axis: x Y-axis: y Data: A curve that passes through the origin (O), increasing from negative infinity to positive infinity, with an inflection point at the origin. The function is labeled f(x)=x³. Context: Illustrates the graphical representation of a basic cubic function, f(x) = x³. **GRAPH**: دالة المقلوب Description: A Cartesian coordinate graph showing a hyperbola with two branches, representing a reciprocal function y=1/x. X-axis: x Y-axis: y Data: A hyperbola with two branches, one in the first quadrant and one in the third quadrant, asymptotic to both the x and y axes. The function is labeled y=1/x, with the condition x≠0. Context: Illustrates the graphical representation of a reciprocal function, y = 1/x. **GRAPH**: دالة الجذر التربيعي Description: A Cartesian coordinate graph showing a curve starting from the origin and extending to the right, representing a square root function y=√x. X-axis: x Y-axis: y Data: A curve that starts at the origin (0,0) and extends into the first quadrant, increasing gradually. The function is labeled y=√x, with the condition x≥0. Key Values: Starting point: (0,0) Context: Illustrates the graphical representation of a basic square root function, y = √x. **GRAPH**: دالة أكبر عدد صحيح Description: A Cartesian coordinate graph showing a step function, representing the greatest integer function f(x)=[x]. X-axis: x Y-axis: y Data: A series of horizontal line segments (steps). Each segment starts with a closed circle on the left and ends with an open circle on the right, indicating that the function value is constant for an interval [n, n+1) and jumps to the next integer at n+1. The function is labeled f(x)=[x]. Context: Illustrates the graphical representation of the greatest integer function, f(x) = [x]. **TABLE**: قيم الدوال المثلثية لبعض الزوايا الخاصة Description: A table listing the sine, cosine, and tangent values for special angles (0, π/6, π/4, π/3, π/2, π). Table Structure: Headers: الزاوية | 0 | π/6 | π/4 | π/3 | π/2 | π Rows: Row 1: sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 | 0 Row 2: cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 | -1 Row 3: tan θ | 0 | √3/3 | 1 | √3 | غير معرف | 0 Calculation needed: No calculations needed, this is a reference table of exact trigonometric values. Context: Provides fundamental trigonometric values for common special angles, essential for trigonometry and calculus.