கணிதம் உங்களுக்காக - 37 - கான்ஸெப்ட்கள்

 

1. General form of the equation of a circle: x² + y² + Dx + Ey + F = 0 (where D, E, and F are constants)

2. Distance between two points in a plane: d = √((x₂ - x₁)² + (y₂ - y₁)²)

3. Midpoint formula: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

4. Slope of a line passing through points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁)

5. Equation of a line in slope-intercept form: y = mx + c (where m is the slope and c is the y-intercept)

6. Equation of a line in point-slope form: y - y₁ = m(x - x₁)

7. General form of a line equation: Ax + By + C = 0

8. Distance from a point to a line: d = |Ax₁ + By₁ + C| / √(A² + B²) (where (x₁, y₁) is the point and Ax + By + C = 0 is the line)

9. Volume of a regular tetrahedron: V = (√2 / 12) a³ (where a is the side length)

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

11. Volume of an octahedron: V = (√2 / 3) a³ (where a is the side length)

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

13. Volume of a regular dodecahedron: V = (15 + 7√5) / 4 a³ (where a is the side length)

14. Surface area of a regular dodecahedron: SA = 3√25 + 10√5 a² (where a is the side length)

15. Volume of a regular icosahedron: V = 5(3 + √5) / 12 a³ (where a is the side length)

16. Surface area of a regular icosahedron: SA = 5√3 a² (where a is the side length)

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

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

19. 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)

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

21. 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)

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

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

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

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

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

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

28. 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)]

29. 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)

30. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

31. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

32. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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