கணிதம் உங்களுக்காக - 24 - கான்ஸெப்ட்கள்

1 . Slope-intercept form of a line: y = mx + c (where m is the slope and c is the y-intercept)

2. Point-slope form of a line: y - y₁ = m(x - x₁) (where m is the slope and (x₁, y₁) is a point on the line)

3. Distance between a point and a line in 2D: d = |Ax₁ + By₁ + C| / √(A² + B²) (where (x₁, y₁) is the point and Ax + By + C = 0 is the line)

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

5. 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)

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

7. Volume of a tetrahedron: V = (1/6)|a · (b × c)| (where a, b, and c are vectors representing the edges)

8. Surface area of a tetrahedron: SA = √3 a² (where a is the side length)

9. Volume of a prism: V = B * h (where B is the area of the base and h is the height)

10. Surface area of a prism: SA = 2B + P * h (where B is the area of the base, P is the perimeter of the base, and h is the height)

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

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

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

14. 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))

15. Power-reduction Formulas:

    - sin²(θ) = (1 - cos(2θ)) / 2

    - cos²(θ) = (1 + cos(2θ)) / 2

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

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

17. Dot product of two vectors: A · B = A₁B₁ + A₂B₂ + A₃B₃

18. Cross product of two vectors: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

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

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

21. 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)

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

23. 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)

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

25. Area of a sector: A = 1/2r²θ (where r is the radius and θ is the central angle in radians)