கணிதம் உங்களுக்காக - 7 - கான்ஸெப்ட்கள்

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

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

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

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

5. Equation of an ellipse (centered at the origin): (x² / a²) + (y² / b²) = 1 (where a is the semi-major axis and b is the semi-minor axis)

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

7. 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)

8. 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)

9. Vector equation of a line: 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)

10. 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)

11. Cross product of two vectors: A × B = |A||B|sin(θ)n (where n is the unit vector perpendicular to the plane containing A and B)

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

13. Volume of a parallelepiped: V = |a · (b × c)| (where a, b, and c are vectors representing the edges)

14. Volume of a frustum of a pyramid: V = (1/3)h(A₁ + A₂ + √(A₁A₂)) (where h is the height, A₁ is the area of the top base, and A₂ is the area of the bottom base)

15. Surface area of a frustum of a pyramid: SA = A₁ + A₂ + (1/2)Pℓ (where P is the perimeter of the base and ℓ is the slant height)

16. Volume of a triangular prism: V = ½bhL (where b is the base length, h is the height, and L is the length of the prism)

17. Surface area of a triangular prism: SA = bh + 2ls + lb (where b is the base length, h is the height, l is the length, and s is the slant height)

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

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

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

21. Segment area: A = ½r²(θ - sinθ)

22. Sum of interior angles of a polygon: (n - 2) × 180° (where n is the number of sides)

23. Sum of exterior angles of a polygon: 360°

24. Number of diagonals in a polygon: n(n - 3) / 2 (where n is the number of sides)

25. Measure of each interior angle of a regular polygon: (n - 2) × 180° / n

26. Measure of each exterior angle of a regular polygon: 360° / n