கணிதம் உங்களுக்காக - 6 - கான்ஸெப்ட்கள்

 1. Volume of a pyramid: V = (1/3) × B × h (where B is the area of the base and h is the height)

2. Surface area of a pyramid: SA = B + ½Pℓ (where B is the area of the base, P is the perimeter of the base, and ℓ is the slant height)

3. Volume of a torus: V = (πr²) × (2πR) = 2π²Rr² (where r is the radius of the tube and R is the distance from the center of the tube to the center of the torus)

4. Surface area of a torus: SA = (2πr) × (2πR) = 4π²Rr

5. Volume of an oblique cylinder: V = πr²h

6. Surface area of an oblique cylinder: SA = 2πrh + 2πr²

7. Sum and Difference Formulas:

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

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

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

8. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

9. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

10. Conversion between polar and Cartesian coordinates:

    - x = r cosθ

    - y = r sinθ

    - r = √(x² + y²)

    - θ = tan⁻¹(y/x)

11. Equation of a circle in polar coordinates: r = R (where R is the radius)

12. Equation of a line in polar coordinates: r = l / cos(θ - φ) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)

13. Dilation: (x, y) → (kx, ky) (where k is the scale factor)

14. Shear transformation: (x, y) → (x + ky, y) (horizontal shear) or (x, y) → (x, y + kx) (vertical shear)

15. Reflection over y = mx + c: (x, y) → ((1 - m²)x + 2my - 2mc) / (1 + m²), (m²x + 2my + 2c) / (1 + m²)

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

17. Surface area of a tetrahedron: SA = √3 a² (for a regular tetrahedron with edge length a)

18. Volume of a parallelepiped: V = |a · (b × c)| (where a, b, and c are vectors representing the edges)

19. Volume of a frustum of a cone: V = (1/3) π h (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)

20. Surface area of a frustum of a cone: SA = π (R₁ + R₂) √((R₁ - R₂)² + h²) + π R₁² + π R₂²