கணிதம் உங்களுக்காக - 22 - கான்ஸெப்ட்கள்

1. Equation of an ellipse in standard form (centered at the origin): (x² / a²) + (y² / b²) = 1

2. Equation of a hyperbola in standard form (centered at the origin): (x² / a²) - (y² / b²) = 1 (horizontal) or (y² / a²) - (x² / b²) = 1 (vertical)

3. Asymptotes of a hyperbola: y = ±(b / a)x (horizontal) or y = ±(a / b)x (vertical)

4. Parametric equations of a circle: 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)

5. Parametric equations of an ellipse: 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)

6. 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)

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

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

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

10. Volume of a spherical sector: V = (2/3)πR²h (where R is the radius of the sphere and h is the height of the spherical sector)

11. Volume of a spherical zone: V = (2/3)πR²(h₂ - h₁) (where R is the radius of the sphere, h₁ is the height of the lower zone, and h₂ is the height of the upper zone)

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

13. Volume of a spherical wedge: V = (2/3)πR²θ (where R is the radius of the sphere and θ is the dihedral angle)

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

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

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

17. 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)

18. Equation of a line in vector form: r = r₀ + t(v) (where r is the position vector, r₀ is a point on the line, v is the direction vector, and t is a scalar)

19. Equation of a plane in vector form: r · n = d (where r is the position vector, n is the normal vector, and d is the distance from the origin)

20. Equation of a sphere in vector form: |r - r₀| = R (where r is the position vector, r₀ is the center, and R is the radius)

21. Heron's formula for the area of a triangle: A = √(s(s - a)(s - b)(s - c)) (where s is the semi-perimeter and a, b, and c are the lengths of the sides)

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

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

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

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

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

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

28. 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)

29. 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)

30. 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)

31. 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)

32. 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)]

33. 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)

34. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

35. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

36. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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