கணிதம் உங்களுக்காக - 26 - கான்ஸெப்ட்கள்

 

1. Equation of a line in standard form: Ax + By + C = 0

2. Distance between two parallel lines: d = |c₂ - c₁| / √(a² + b²) (where the equations of the lines are Ax + By + C₁ = 0 and Ax + By + C₂ = 0)

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

4. Slope of the perpendicular bisector of a line segment: m = -1/m (where m is the slope of the original line)

5. Equation of the perpendicular bisector of a line segment: y - y₁ = -1/m(x - x₁) (where m is the slope of the original line and (x₁, y₁) is the midpoint of the line segment)

6. Volume of a truncated cone (frustum of a cone): 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)

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

8. Volume of a frustum of a pyramid: V = (1/3)h(A₁ + A₂ + √(A₁A₂)) (where h is the height, A₁ is the area of the top base, and A₂ is the area of the bottom base)

9. Surface area of a frustum of a pyramid: SA = A₁ + A₂ + (1/2)Pℓ (where P is the perimeter of the base and ℓ is the slant height)

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

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

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

13. Segment area: A = 1/2r²(θ - sinθ)

14. 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)

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

16. 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)

17. Length of a median of a triangle: m = ½ √(2b² + 2c² - a²) (where a, b, and c are the lengths of the sides of the triangle, and m is the median to side a)

18. Length of an altitude in a triangle: h = 2A / a (where A is the area of the triangle and a is the base)

19. 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)

20. 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)

21. 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)

22. 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)

23. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

24. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((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