கணிதம் உங்களுக்காக - 30 - கான்ஸெப்ட்கள்

 

1. Volume of a cylindrical shell: V = 2πrh × (R - r) (where R is the outer radius and r is the inner radius)

2. Surface area of a cylindrical shell: SA = 2πh(R + r) (where R is the outer radius and r is the inner radius)

3. Volume of a truncated cone (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)

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

5. 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)

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

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

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

9. 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)

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

11. 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)

12. 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)]

13. 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)

14. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

15. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

16. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

17. 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)

18. 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)

19. 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)

20. 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)

21. 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)

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

23. 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)

24. 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)

25. 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)

26. 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)

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

2. 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)

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

4. Volume of a spherical lune: V = (1/3)πR²θ (where R is the radius of the sphere and θ is the dihedral angle)

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

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

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

8. 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)

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

10. 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)

11. 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)]

12. 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)

13. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

14. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

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

    - tan(2A) = 2 tanA / (1 - tan²A)

15. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

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

16. Dot product: A · B = A₁B₁ + A₂B₂ + A₃B₃

17. Vector cross product: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

18. Magnitude of a vector in 3D: |A| = √(A₁² + A₂² + A₃²)

19. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

20. Vector projection of vector A onto vector B: Proj_B(A) = ((A · B) / |B|²) B

21. Angle between two vectors: cos(θ) = (A · B) / (|A||B|)

22. 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)

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

24. 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)

25. 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)

26. 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)

27. 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)

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

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

3. 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)

4. 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)

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

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

7. 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)

8. 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)

9. 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)

10. 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)

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

12. 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)

13. 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)

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

15. 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)

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

17. Volume of an oblique prism: V = B * h (where B is the area of the base and h is the height)

18. Surface area of an oblique prism: SA = 2B + P * h (where B is the area of the base, P is the perimeter of the base, and h is the height)

19. Volume of a truncated cylinder: V = πh(r₁² + r₁r₂ + r₂²) / 3 (where h is the height, r₁ is the radius of the top base, and r₂ is the radius of the bottom base)

20. Surface area of a truncated cylinder: SA = π(r₁ + r₂)√((r₂ - r₁)² + h²) + πr₁² + πr₂²

21. 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)

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

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

24. 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)

25. Rotation about the origin by θ degrees: (x', y') = (x cosθ - y sinθ, x sinθ + y cosθ)

26. Translation: (x, y) → (x + a, y + b)

27. Reflection over the x-axis: (x, y) → (x, -y)

28. Reflection over the y-axis: (x, y) → (-x, y)

29. Reflection over the line y = x: (x, y) → (y, x)

30. Reflection over the line y = -x: (x, y) → (-y, -x)

31. Dilation with respect to the origin: (x, y) → (kx, ky) (where k is the scale factor)

32. Dilation with respect to a point (h, k): (x', y') = (kx + (1 - k)h, ky + (1 - k)k)

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