கணிதம் உங்களுக்காக - 27 - கான்ஸெப்ட்கள்

 

1. 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)

2. 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)

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

4. Volume of a prismoid: V = (h/6)(A₁ + 4Aₘ + A₂) (where A₁ and A₂ are the areas of the parallel bases, Aₘ is the area of the midsection, and h is the height)

5. Surface area of a prismoid: SA = 2Aₘ + hP (where Aₘ is the area of the midsection and P is the perimeter of the base)

6. Euler's formula for polyhedra: V - E + F = 2 (where V is the number of vertices, E is the number of edges, and F is the number of faces)

7. Surface area of a cylinder: SA = 2πr(h + r)

8. Volume of a cylinder: V = πr²h

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

10. Volume of a cone: V = (1/3)πr²h

11. Surface area of a sphere: SA = 4πr²

12. Volume of a sphere: V = (4/3)πr³

13. 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)

14. 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)

15. 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)

16. Distance between two parallel planes: d = |d₁ - d₂| / |n| (where d₁ and d₂ are the distances from the origin and n is the normal vector)

17. 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)

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

19. 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)

20. 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)]

21. 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)

22. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

23. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

24. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

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

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

3. Volume of a prismoid: V = (h/6)(A₁ + 4Aₘ + A₂) (where A₁ and A₂ are the areas of the parallel bases, Aₘ is the area of the midsection, and h is the height)

4. 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)

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

6. 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)

7. Length of a median of a triangle: m = ½ √(2b² + 2c² - a²) (where a, b, and c are the lengths of the sides of the triangle, and m is the median to side a)

8. Length of an altitude in a triangle: h = 2A / a (where A is the area of the triangle and a is the base)

9. Length of an angle bisector in a triangle: l = √(bc[1 - (a² / (b + c)²)]) (where a, b, and c are the lengths of the sides of the triangle, and l is the angle bisector)

10. Brahmagupta's formula (area of a cyclic quadrilateral): A = √((s - a)(s - b)(s - c)(s - d) - abcd cos²(½θ)) (where θ is the sum of the opposite angles)

11. Area of a cyclic quadrilateral: A = √((s - a)(s - b)(s - c)(s - d)) (where a, b, c, and d are the lengths of the sides, and s is the semi-perimeter)

12. Area of an inscribed circle in a triangle: A = r × s (where r is the radius of the inscribed circle and s is the semi-perimeter)

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

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

15. 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)

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

17. 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)

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. Distance between two parallel planes: d = |d₁ - d₂| / |n| (where d₁ and d₂ are the distances from the origin and n is the normal vector)

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)

24. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

25. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

26. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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