கணிதம் உங்களுக்காக - 15 - கான்ஸெப்ட்கள்

 1. 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)]

2. 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)

3. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

4. Area of a rhombus: A = ½(d₁ × d₂) (where d₁ and d₂ are the lengths of the diagonals)

5. Perimeter of a rhombus: P = 4a (where a is the side length)

6. Area of a kite: A = ½(d₁ × d₂) (where d₁ and d₂ are the lengths of the diagonals)

7. Perimeter of a kite: P = 2(a + b) (where a and b are the lengths of the pairs of equal sides)

8. Area of a trapezoid: A = ½(a + b)h (where a and b are the lengths of the parallel sides and h is the height)

9. Perimeter of a trapezoid: P = a + b + c + d (where a, b, c, and d are the lengths of the sides)

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

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

12. 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)

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

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

15. 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)

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

17. 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)

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. Segment area: A = ½r²(θ - sinθ)

21. Sum of the exterior angles of a polygon: 360°

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

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

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

25. Area of a regular polygon: A = ½ × Perimeter × Apothem (where Perimeter = n × s, n is the number of sides, s is the side length, and the Apothem is the perpendicular distance from the center to a side)