கணிதம் உங்களுக்காக - 21 - கான்ஸெப்ட்கள்

1. Volume of an oblique cylinder: V = πr²h

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

3. Volume of an oblique cone: V = (1/3)πr²h

4. Surface area of an oblique cone: SA = πr(r + √(r² + h²))

5. Volume of a spherical cap: V = (1/3)πh²(3R - h) (where h is the height of the cap and R is the radius of the sphere)

6. Surface area of a spherical cap: SA = 2πRh (where h is the height of the cap and R is the radius of the sphere)

7. Volume of a spherical segment: V = (1/6)πh(3a² + 3b² + h²) (where h is the height of the segment, a and b are the radii of the segment's bases)

8. Volume of a paraboloid: V = (1/2)πr²h

9. Rotation by 90 degrees: (x', y') = (-y, x)

10. Rotation by 180 degrees: (x', y') = (-x, -y)

11. Rotation by 270 degrees: (x', y') = (y, -x)

12. Reflection over y = x: (x', y') = (y, x)

13. Reflection over y = -x: (x', y') = (-y, -x)

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

15. Dilation with respect to a point: (x', y') = (kx + (1 - k)x₀, ky + (1 - k)y₀) (where (x₀, y₀) is the point of dilation and k is the scale factor)

16. Reflection over a line: (x', y') = ((1 - m²)x + 2my - 2mc) / (1 + m²), (y(1 - m²) + 2mx + 2c) / (1 + m²) (for the line y = mx + c)

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

18. Equation of a hyperbola: (x² / a²) - (y² / b²) = 1 (horizontal) or (y² / a²) - (x² / b²) = 1 (vertical)

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

20. 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)

21. 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)

22. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

23. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

24. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

25. Magnitude of a vector: |A| = √(A₁² + A₂² + A₃²)

26. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

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

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

29. Perimeter of a parallelogram: P = 2(a + b) (where a and b are the lengths of the sides)

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

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

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

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

34. Area of a regular hexagon: A = (3√3 / 2) s² (where s is the side length)

35. Perimeter of a regular hexagon: P = 6s (where s is the side length)

36. Area of a regular pentagon: A = (1/4)√(5(5 + 2√5)) s² (where s is the side length)

37. Perimeter of a regular pentagon: P = 5s (where s is the side length)

1. Volume of a regular tetrahedron: V = (√2 / 12) a³ (where a is the side length)

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

3. Volume of a truncated cube: V = (a³ / 3) (where a is the side length of the original cube and b is the side length of the truncated portion)

4. Surface area of a truncated cube: SA = 3a² + 2√3a² (where a is the side length of the original cube)

5. Volume of a cylindrical shell: V = 2πRh (R - r) (where R is the outer radius, r is the inner radius, and h is the height)

6. Surface area of a cylindrical shell: SA = 2πh(R + r) (where R is the outer radius, r is the inner radius, and h is the height)

7. Conversion from Cartesian to polar coordinates: r = √(x² + y²), θ = atan2(y, x)

8. Equation of a line in polar coordinates: r(θ) = (l) / (cos(θ - φ)) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)

9. Equation of a circle in polar coordinates: r(θ) = R (where R is the radius)

10. 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)

11. Triple angle formulas:

    - sin(3θ) = 3 sin(θ) - 4 sin³(θ)

    - cos(3θ) = 4 cos(θ) - 3 cos³(θ)

    - tan(3θ) = (3 tan(θ) - tan³(θ)) / (1 - 3 tan²(θ))

12. Power-reduction formulas:

    - sin²(θ) = (1 - cos(2θ)) / 2

    - cos²(θ) = (1 + cos(2θ)) / 2

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

13. 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)]

14. 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)

15. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

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

16. Central angle theorem: The measure of the central angle is equal to the measure of the intercepted arc.

17. Inscribed angle theorem: The measure of an inscribed angle is half the measure of its intercepted arc.

18. Angle between a tangent and a chord: The measure of the angle formed by a tangent and a chord is half the measure of the intercepted arc.

19. Alternate segment theorem: The angle between the tangent and a chord through the point of contact is equal to the angle in the alternate segment.

20. 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)

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

22. 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)

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

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

25. Segment area: A = ½r²(θ - sinθ)

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26. Dot product: A · B = A₁B₁ + A₂B₂ + A₃B₃

27. Vector cross product: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

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

29. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

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

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

32. Length of a median of a triangle: m = ½ √(2b² + 2c² - a²) (where a, b, and c are the lengths of the sides of the triangle, and m is the median to side a)

33. Length of an altitude in a triangle: h = 2A / a (where A is the area of the triangle and a is the base)

34. Length of an angle bisector in a triangle: l = √(bc[1 - (a² / (b + c)²)]) (where a, b, and c are the lengths of the sides of the triangle, and l is the angle bisector)

35. Area of a cyclic quadrilateral: A = √((s - a)(s - b)(s - c)(s - d)) (where a, b, c, and d are the lengths of the sides, and s is the semi-perimeter)

36. Brahmagupta's formula (area of a cyclic quadrilateral): A = √((s - a)(s - b)(s - c)(s - d) - abcd cos²(½θ)) (where θ is the sum of the opposite angles)

37. Area of an inscribed circle in a triangle: A = r × s (where r is the radius of the inscribed circle and s is the semi-perimeter)