கணிதம் உங்களுக்காக - 34 - கான்ஸெப்ட்கள்

1. Volume of a right circular cone: V = (1/3)πr²h (where r is the radius of the base and h is the height)

2. Surface area of a right circular cone: SA = πr(r + √(r² + h²))

3. Volume of a right circular cylinder: V = πr²h (where r is the radius of the base and h is the height)

4. Surface area of a right circular cylinder: SA = 2πrh + 2πr²

5. Volume of a spherical segment: V = (1/6)πh(3a² + 3b² + h²) (where h is the height of the segment, a and b are the radii of the segment's bases)

6. Surface area of a spherical segment: SA = 2πRh (where R is the radius of the sphere and h is the height of the segment)

7. Length of the altitude of a right triangle: h = (a * b) / c (where a and b are the legs and c is the hypotenuse)

8. Length of the altitude of an equilateral triangle: h = (√3 / 2) * a (where a is the side length)

9. Radius of the circumscribed circle of a triangle: R = (abc) / (4A) (where a, b, and c are the side lengths and A is the area of the triangle)

10. Radius of the inscribed circle of a triangle: r = A / s (where A is the area of the triangle and s is the semi-perimeter)

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

12. Length of the diagonal of a rectangle: d = √(a² + b²) (where a and b are the side lengths)

13. 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)

14. Length of the side of a regular polygon: s = 2R sin(π / n) (where R is the radius of the circumscribed circle and n is the number of sides)

15. 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)

16. 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)

17. Equation of the parabola (vertex form): y - k = a(x - h)² (where (h, k) is the vertex)

18. Equation of the ellipse (standard form): (x - h)² / a² + (y - k)² / b² = 1 (where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis)

19. Equation of the hyperbola (standard form): (x - h)² / a² - (y - k)² / b² = 1 (horizontal) or (y - k)² / a² - (x - h)² / b² = 1 (vertical)

20. Slopes of the asymptotes of a hyperbola: ±(b / a) (horizontal) or ±(a / b) (vertical)

21. Distance between two parallel lines: d = |c₂ - c₁| / √(a² + b²) (where the equations of the lines are Ax + By + C₁ = 0 and Ax + By + C₂ = 0)

22. Distance between a point and a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

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

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

25. Volume of a parallelepiped: V = |A · (B × C)| (where A, B, and C are vectors)

26. 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)

27. 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)]

28. 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)

29. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

30. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

31. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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