கணிதம் உங்களுக்காக - 31 - கான்ஸெப்ட்கள்

 1. Angle between two lines: tan(θ) = |(m₁ - m₂) / (1 + m₁m₂)| (where m₁ and m₂ are the slopes of the lines)

2. Area of a triangle (using coordinates): A = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

3. Centroid of a triangle (using coordinates): G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)

4. Circumcenter of a triangle (using coordinates): Intersection of perpendicular bisectors of the sides of the triangle

5. Incenter of a triangle (using coordinates): Intersection of angle bisectors of the triangle

6. Volume of a regular polyhedron: V = (1/6) a³√(2/3)[5n(3 + 2√5)] (where a is the side length and n is the number of faces)

7. Surface area of a regular polyhedron: SA = n a²√3 (where a is the side length and n is the number of faces)

8. Volume of a truncated cube: V = (a³ / 3) (where a is the side length of the original cube and b is the side length of the truncated portion)

9. Surface area of a truncated cube: SA = 3a² + 2√3a² (where a is the side length of the original cube)

10. Volume of a cylindrical shell: V = 2πRh (R - r) (where R is the outer radius, r is the inner radius, and h is the height)

11. Surface area of a cylindrical shell: SA = 2πh(R + r) (where R is the outer radius, r is the inner radius, and h is the height)

12. Rotation around the origin by θ degrees: (x', y') = (x cosθ - y sinθ, x sinθ + y cosθ)

13. Reflection over a line y = mx + c: (x', y') = ((x(1 - m²) + 2my - 2mc) / (1 + m²), (y(1 - m²) + 2mx + 2c) / (1 + m²))

14. Dilation with respect to the origin: (x, y) → (kx, ky) (where k is the scale factor)

15. Horizontal shear transformation: (x', y') = (x + ky, y)

16. Vertical shear transformation: (x', y') = (x, y + kx)

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

18. Area of a sector: A = ½r²θ (where r is the radius and θ is the central angle in radians)

19. Segment area: A = ½r²(θ - sinθ)

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

21. 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)

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

23. 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)

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

25. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

26. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

27. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

28. Dot product: A · B = A₁B₁ + A₂B₂ + A₃B₃

29. Vector cross product: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

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

31. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

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

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

34. 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)

35. Volume of a parallelepiped: V = |A · (B × C)| (where A, B, and C are vectors)

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

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