கணிதம் உங்களுக்காக - 28 - கான்ஸெப்ட்கள்

 

1. Equation of the parabola (general form): Ax² + Bxy + Cy² + Dx + Ey + F = 0

2. Equation of the hyperbola (centered at the origin): (x² / a²) - (y² / b²) = 1 (horizontal) or (y² / a²) - (x² / b²) = 1 (vertical)

3. Equation of the ellipse (centered at the origin): (x² / a²) + (y² / b²) = 1

4. Length of a circle's arc using radians: L = rθ (where r is the radius and θ is the central angle in radians)

5. Length of a circle's arc using degrees: L = (θ / 360) × 2πr

6. Length of the tangent from an external point to a circle: PT = √(d² - r²) (where d is the distance from the external point to the circle's center, and r is the radius)

7. Volume of an irregular tetrahedron: V = (1/6) |AB · (AC × AD)| (where AB, AC, and AD are vectors representing the edges)

8. Surface area of a regular octahedron: SA = 2√3a² (where a is the side length)

9. Volume of a truncated 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)

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

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

12. 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)

13. Addition and Subtraction Formulas:

    - sin(A ± B) = sinA cosB ± cosA sinB

    - cos(A ± B) = cosA cosB ∓ sinA sinB

    - tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

14. Multiple Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

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

16. Equation of a circle (standard form): (x - h)² + (y - k)² = r² (where (h, k) is the center and r is the radius)

17. Equation of a circle (general form): x² + y² + Dx + Ey + F = 0 (where D, E, and F are constants)

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

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

20. Segment area: A = ½r²(θ - sinθ)

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

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

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