கணிதம் உங்களுக்காக - 18 - கான்ஸெப்ட்கள்

 1. Angle between two lines: tan(θ) = |(m₁ - m₂) / (1 + m₁m₂)| (where m₁ and m₂ are the slopes of the lines)

2. Area of a triangle (using coordinates): A = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

3. Centroid of a triangle (using coordinates): G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)

4. Circumcenter of a triangle (using coordinates): Intersection of perpendicular bisectors of the sides of the triangle

5. Incenter of a triangle (using coordinates): Intersection of angle bisectors of the triangle

6. Volume of an oblique cylinder: V = πr²h

7. Surface area of an oblique cylinder: SA = 2πrh + 2πr²

8. Volume of a regular dodecahedron: V = (15 + 7√5) / 4 a³ (where a is the side length)

9. Surface area of a regular dodecahedron: SA = 3√25 + 10√5 a² (where a is the side length)

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

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

12. Power-reduction Formulas:

    - sin²(A) = (1 - cos(2A)) / 2

    - cos²(A) = (1 + cos(2A)) / 2

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

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. Equation of a circle (standard form): (x - h)² + (y - k)² = r² (where (h, k) is the center and r is the radius)

15. Equation of a circle (general form): x² + y² + Dx + Ey + F = 0 (where D, E, and F are constants)

16. Equation of a parabola (standard form): y = ax² + bx + c or (y - k) = a(x - h)² (where (h, k) is the vertex)

17. Equation of an 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)

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

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

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

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

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

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

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

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

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

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

28. Length of a chord: c = 2r sin(θ/2) (where r is the radius and θ is the central angle in radians)

29. Segment area: A = ½r²(θ - sinθ)

30. 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)

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

32. 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)