கணிதம் உங்களுக்காக - 11 - கான்ஸெப்ட்கள்

 1. Length of a chord of a circle: c = 2r sin(θ/2) (where r is the radius and θ is the central angle in radians)

2. Length of a tangent segment: t = r√(2 - 2cos(θ)) (where r is the radius and θ is the central angle in radians)

3. Length of an arc of a circle: L = rθ (where r is the radius and θ is the central angle in radians)

4. Distance from a point to a line in 3D: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

5. Distance between two points in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

6. Midpoint in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)

7. Volume of a tetrahedron: V = (1/6)|a · (b × c)| (where a, b, and c are vectors representing the edges)

8. Surface area of a tetrahedron: SA = √3 a² (where a is the side length)

9. Volume of a torus: V = 2π²Rr² (where R is the distance from the center of the tube to the center of the torus, and r is the radius of the tube)

10. Surface area of a torus: SA = 4π²Rr (where R is the distance from the center of the tube to the center of the torus, and r is the radius of the tube)

11. Volume of a frustum: V = (1/3)πh(R₁² + R₁R₂ + R₂²) (where h is the height, R₁ is the radius of the top base, and R₂ is the radius of the bottom base)

12. Surface area of a frustum: SA = π(R₁ + R₂)√((R₁ - R₂)² + h²) + πR₁² + πR₂²

13. Polar to Cartesian coordinates: x = r cos(θ), y = r sin(θ)

14. Cartesian to Polar coordinates: r = √(x² + y²), θ = tan⁻¹(y / x)

15. Polar form of the equation of a line: r = l / cos(θ - φ) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)

16. Polar form of the equation of a circle: r(θ) = R (where R is the radius)

17. Parametric equations for a circle in polar coordinates: x = R cos(θ), y = R sin(θ) (where R is the radius and θ is the parameter)

18. Law of Tangents: (a - b) / (a + b) = tan[(A - B)/2] / tan[(A + B)/2]

19. Law of Cotangents: (a - b) / (a + b) = cot[(A - B)/2] / cot[(A + B)/2]

20. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

21. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

22. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

23. Scalar triple product: V = A · (B × C) (where A, B, and C are vectors)

24. Vector triple product: A × (B × C) = (A · C)B - (A · B)C

25. Distance between two skew lines: d = |(A₁ - A₂) · (B₁ × B₂)| / |B₁ × B₂| (where A₁ and A₂ are points on the lines, and B₁ and B₂ are direction vectors)

26. Magnitude of a vector in 3D: |A| = √(A₁² + A₂² + A₃²)

27. Dot product of two vectors: A · B = A₁B₁ + A₂B₂ + A₃B₃

28. Cross product of two vectors: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

29. Angle between two vectors: cos(θ) = (A · B) / (|A||B|)

30. Vector projection of A onto B: Proj_B(A) = ((A · B) / |B|²) B

31. Power of a Point Theorem: If a point P lies outside a circle and two secant lines PA and PB intersect the circle at A and B, then PA × PB = PC × PD (where C and D are points of intersection)

32. Segment lengths in a circle: If two chords AB and CD intersect at point P inside the circle, then PA × PB = PC × PD.

33. Length of a tangent segment: If a tangent from an external point P touches the circle at T, then PT² = PA × PB (where A and B are points of intersection of a secant through P)

34. Length of an arc: L = rθ (where r is the radius and θ is the central angle in radians)

35. Area of a sector: A = 1/2r²θ (where r is the radius and θ is the central angle in radians)

36. Segment area: A = 1/2r²(θ - sinθ)

37. Length of the altitude of a right triangle: h = (a * b) / c (where a and b are the legs and c is the hypotenuse)

38. Radius of the circumscribed circle of a triangle: R = (abc) / (4A) (where a, b, and c are the side lengths and A is the area of the triangle)

39. Length of the median of a trapezoid: m = (a + b) / 2 (where a and b are the lengths of the parallel sides)

40. Length of the diagonal of a rectangle: d = √(a² + b²) (where a and b are the side lengths)

41. Length of the diagonal of a parallelogram: d = √(a² + b² + 2ab cos(θ)) (where a and b are the side lengths and θ is the angle between the sides)

42. Heron's formula for the area of a triangle: A = √(s(s - a)(s - b)(s - c)) (where s is the semi-perimeter and a, b, and c are the lengths of the sides)

43. Length of an angle bisector in a triangle: l = √(bc[1 - (a² / (b + c)²)]) (where a, b, and c are the lengths of the sides of the triangle, and l is the angle bisector)

44. Area of a cyclic quadrilateral: A = √((s - a)(s - b)(s - c)(s - d)) (where a, b, c, and d are the lengths of the sides, and s is the semi-perimeter)

45. Brahmagupta's formula (area of a cyclic quadrilateral): A = √((s - a)(s - b)(s - c)(s - d) - abcd cos²(½θ)) (where θ is the sum of the opposite angles)

46. Area of an inscribed circle in a triangle: A = r × s (where r is the radius of the inscribed circle and s is the semi-perimeter)

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