கணிதம் உங்களுக்காக - 32 - கான்ஸெப்ட்கள்

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

2. Equation of a circle in parametric form: x = h + r cos(t), y = k + r sin(t) (where (h, k) is the center, r is the radius, and t is the parameter)

3. Equation of an ellipse in parametric form: x = h + a cos(t), y = k + b sin(t) (where (h, k) is the center, a is the semi-major axis, b is the semi-minor axis, and t is the parameter)

4. Equation of a hyperbola in parametric form: x = h + a sec(t), y = k + b tan(t) (where (h, k) is the center, a is the semi-major axis, b is the semi-minor axis, and t is the parameter)

5. Locus of a point: The set of all points that satisfy a given condition.

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

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

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

11. Surface area of a parallelepiped: SA = 2(ab + bc + ca) (where a, b, and c are the lengths of the edges)

12. Volume of an oblique cylinder: V = πr²h

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

14. Volume of a cone: V = 1/3πr²h

15. Surface area of a cone: SA = πr(r + l) (where l is the slant height)

16. Length of the diagonal of a regular polygon: d = √(2r²(1 - cos(2π/n))) (where r is the radius of the circumscribed circle and n is the number of sides)

17. Radius of the circumscribed circle of a regular polygon: R = (s / 2) * csc(π / n) (where s is the side length and n is the number of sides)

18. Radius of the inscribed circle of a regular polygon: r = (s / 2) * cot(π / n) (where s is the side length and n is the number of sides)

19. Length of a median of a trapezoid: m = (a + b) / 2 (where a and b are the lengths of the parallel sides)

20. Length of the diagonal of a parallelogram: d = √(a² + b² + 2ab cos(θ)) (where a and b are the side lengths and θ is the angle between the sides)

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

22. Law of Cotangents: (a - b) / (a + b) = cot[(A - B)/2] / cot[(A + B)/2]

23. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

24. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

25. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

26. Scalar triple product: V = A · (B × C) (where A, B, and C are vectors)

27. Vector triple product: A × (B × C) = (A · C)B - (A · B)C

28. 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)