கணிதம் உங்களுக்காக - 33 - கான்ஸெப்ட்கள்

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

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

3. Distance between two points (x₁, y₁) and (x₂, y₂): d = √((x₂ - x₁)² + (y₂ - y₁)²)

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

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

6. Distance from a point to a line Ax + By + C = 0: d = |Ax₁ + By₁ + C| / √(A² + B²)

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

8. Equation of a parabola in standard form: y = ax² + bx + c or (y - k) = a(x - h)² (where (h, k) is the vertex)

9. Volume of a 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)

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

11. Volume of an ellipsoid: V = (4/3)πabc (where a, b, and c are the semi-axes)

12. Surface area of an ellipsoid (approximate): SA = 4π[(a²b^(p) + a²c^(p) + b²c^(p))/3]^(1/p) (where p ≈ 1.6075)

13. Volume of a paraboloid: V = (1/2)πr²h

14. Volume of a truncated 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)

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

16. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

17. Law of Cosines: c² = a² + b² - 2ab cos(C)

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

19. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

20. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

21. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

22. 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)]

23. 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)