Wednesday, February 19, 2025

கணிதம் உங்களுக்காக - 8 - கான்ஸெப்ட்கள்

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

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

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

4. Perimeter of a regular polygon: P = n × s (where n is the number of sides and s is the side length)

5. Distance from a point to a line: d = |Ax₁ + By₁ + C| / √(A² + B²) (where (x₁, y₁) is the point and Ax + By + C = 0 is the line)
6. Equation of a circle with center (h, k) and radius r: (x - h)² + (y - k)² = r²
7. Equation of an ellipse with center (h, k): ((x - h)² / a²) + ((y - k)² / b²) = 1 (where a is the semi-major axis and b is the semi-minor axis)
8. Equation of a hyperbola with center (h, k): ((x - h)² / a²) - ((y - k)² / b²) = 1 (horizontal) or ((y - k)² / a²) - ((x - h)² / b²) = 1 (vertical)

9. Magnitude of a vector: |A| = √(A₁² + A₂² + A₃²)
10. Direction cosines of a vector: cosα = A₁ / |A|, cosβ = A₂ / |A|, cosγ = A₃ / |A|
11. Scalar projection of A onto B: proj_B(A) = (A · B) / |B|
12. Vector projection of A onto B: Proj_B(A) = ((A · B) / |B|²) B

13. Rotation by θ degrees about the origin: (x', y') = (x cosθ - y sinθ, x sinθ + y cosθ)
14. Reflection over the line y = mx + c: (x, y) → (x(1 - m²) + 2my - 2mc) / (1 + m²), y(1 - m²) + 2mx + 2c) / (1 + m²)
15. Dilation with center (h, k) and scale factor k: (x', y') = (h + k(x - h), k + k(y - k))

16. Sum of interior angles of a polygon: (n - 2) × 180° (where n is the number of sides)
17. Sum of exterior angles of a polygon: 360°
18. Interior angle of a regular polygon: (n - 2) × 180° / n
19. Exterior angle of a regular polygon: 360° / n
20. Area of a regular polygon: A = (1/4)n s² cot(π/n) (where n is the number of sides and s is the side length)

16. Sum of interior angles of a polygon: (n - 2) × 180° (where n is the number of sides)
17. Sum of exterior angles of a polygon: 360°
18. Interior angle of a regular polygon: (n - 2) × 180° / n
19. Exterior angle of a regular polygon: 360° / n
20. Area of a regular polygon: A = (1/4)n s² cot(π/n) (where n is the number of sides and s is the side length)

21. Law of Tangents: (a - b) / (a + b) = (tan((A - B)/2)) / (tan((A + B)/2))
22. Product-to-Sum Formulas:
    - sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]
    - cos(A)cos(B) = 1/2[cos(A + B) + cos(A - B)]
    - sin(A)cos(B) = 1/2[sin(A + B) + sin(A - B)]
23. Half-Angle Formulas:
    - sin²(A/2) = (1 - cosA) / 2
    - cos²(A/2) = (1 + cosA) / 2
    - tan²(A/2) = (1 - cosA) / (1 + cosA)
24. Sum and Difference Formulas:
    - sin(A ± B) = sinA cosB ± cosA sinB
    - cos(A ± B) = cosA cosB ∓ sinA sinB
    - tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
25. Double Angle Formulas:
    - sin(2A) = 2 sinA cosA
    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A
    - tan(2A) = 2 tanA / (1 - tan²A)

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