கணிதம் உங்களுக்காக - 21 - கான்ஸெப்ட்கள்

1. Volume of an ellipsoid: V = (4/3)πabc (where a, b, and c are the semi-axes)

2. Surface area of an ellipsoid (approximate): SA = 4π[(a²b^(p) + a²c^(p) + b²c^(p))/3]^(1/p) (where p ≈ 1.6075)

3. Volume of a regular polyhedron: V = (1/6) a³√(2/3)[5n(3 + 2√5)] (where a is the side length and n is the number of faces)

4. Surface area of a regular polyhedron: SA = n a²√3 (where a is the side length and n is the number of faces)

5. Rotation around an arbitrary point: (x', y') = ((x - x₀) cosθ - (y - y₀) sinθ + x₀, (x - x₀) sinθ + (y - y₀) cosθ + y₀) (where (x₀, y₀) is the point)

6. Reflection over y = mx + c: (x', y') = ((x(1 - m²) + 2my - 2mc) / (1 + m²), (y(1 - m²) + 2mx + 2c) / (1 + m²))

7. Dilation with respect to a point: (x', y') = (kx + (1 - k)x₀, ky + (1 - k)y₀) (where (x₀, y₀) is the point of dilation and k is the scale factor)

8. Conversion from Cartesian to polar coordinates: r = √(x² + y²), θ = atan2(y, x)

9. Equation of a line in polar coordinates: r(θ) = (l) / (cos(θ - φ)) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)

10. Equation of a circle in polar coordinates: r(θ) = R (where R is the radius)

11. Triple angle formulas:

    - sin(3θ) = 3 sin(θ) - 4 sin³(θ)

    - cos(3θ) = 4 cos³(θ) - 3 cos(θ)

    - tan(3θ) = (3 tan(θ) - tan³(θ)) / (1 - 3 tan²(θ))

12. Power-reduction formulas:

    - sin²(θ) = (1 - cos(2θ)) / 2

    - cos²(θ) = (1 + cos(2θ)) / 2

    - tan²(θ) = (1 - cos(2θ)) / (1 + cos(2θ))

13. Central angle theorem: The measure of the central angle is equal to the measure of the intercepted arc.

14. Inscribed angle theorem: The measure of an inscribed angle is half the measure of its intercepted arc.

15. Angle between a tangent and a chord: The measure of the angle formed by a tangent and a chord is half the measure of the intercepted arc.

16. Alternate segment theorem: The angle between the tangent and a chord through the point of contact is equal to the angle in the alternate segment.

17. Area of a parallelogram: A = b × h (where b is the base and h is the height)

18. Perimeter of a parallelogram: P = 2(a + b) (where a and b are the lengths of the sides)

19. Area of a regular hexagon: A = (3√3 / 2) s² (where s is the side length)

20. Perimeter of a regular hexagon: P = 6s (where s is the side length)

21. Area of a regular pentagon: A = (1/4)√(5(5 + 2√5)) s² (where s is the side length)

22. Perimeter of a regular pentagon: P = 5s (where s is the side length)

1. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

2. Law of Cosines: c² = a² + b² - 2ab cos(C)

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

4. Area of a triangle using Heron's formula: 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)

5. Area of a triangle using trigonometry: A = ½ab sin(C)

6. Volume of a torus: V = (πr²) × (2πR) = 2π²Rr² (where r is the radius of the tube and R is the distance from the center of the tube to the center of the torus)

7. Surface area of a torus: SA = (2πr) × (2πR) = 4π²Rr

8. Volume of a 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)

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

10. Central angle theorem: The measure of the central angle is equal to the measure of the intercepted arc.

11. Inscribed angle theorem: The measure of an inscribed angle is half the measure of its intercepted arc.

12. Angle between a tangent and a chord: The measure of the angle formed by a tangent and a chord is half the measure of the intercepted arc.

13. Alternate segment theorem: The angle between the tangent and a chord through the point of contact is equal to the angle in the alternate segment.

14. Power of a Point Theorem: If a point P lies outside a circle and two secant lines PA and PB intersect the circle at A and B, then PA × PB = PC × PD (where C and D are points of intersection)

15. Segment lengths in a circle: If two chords AB and CD intersect at point P inside the circle, then PA × PB = PC × PD.

16. Length of a tangent segment: If a tangent from an external point P touches the circle at T, then PT² = PA × PB (where A and B are points of intersection of a secant through P)

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

18. Area of a sector: A = ½r²θ (where r is the radius and θ is the central angle in radians)

19. Segment area: A = ½r²(θ - sinθ)

20. Distance formula in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

21. Midpoint formula in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)

22. Equation of a sphere: (x - h)² + (y - k)² + (z - l)² = r² (where (h, k, l) is the center and r is the radius)

23. Equation of a plane: Ax + By + Cz + D = 0

24. Distance from a point to a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

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

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

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

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

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

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

31. 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)

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

33. 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)

34. 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)

35. 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)

36. 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)

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

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

39. Equation of a circle in polar coordinates: r = R (where R is the radius)

40. Equation of a line in polar coordinates: r = l / cos(θ - φ) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)