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 an octahedron: V = (√2 / 3) a³ (where a is the side length)
4. Surface area of an octahedron: SA = 2√3 a² (where a is the side length)
5. Volume of a dodecahedron: V = (15 + 7√5) / 4 a³ (where a is the side length)
6. Surface area of a dodecahedron: SA = 3√25 + 10√5 a² (where a is the side length)
7. Volume of an icosahedron: V = 5(3 + √5) / 12 a³ (where a is the side length)
8. Surface area of an icosahedron: SA = 5√3 a² (where a is the side length)
9. In any triangle (not necessarily right): a² = b² + c² - 2bc cos(A) (where A is the angle opposite side a)
10. Law of Cosines: c² = a² + b² - 2ab cos(C) (where C is the angle opposite side c)
11. 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)
12. Double Angle Formulas:
- sin(2A) = 2 sinA cosA
- cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A
- tan(2A) = 2 tanA / (1 - tan²A)
13. Half-Angle Formulas:
- sin²(A/2) = (1 - cosA) / 2
- cos²(A/2) = (1 + cosA) / 2
- tan²(A/2) = (1 - cosA) / (1 + cosA)
14. 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)]
15. Standard form of a parabola (vertical): y - k = a(x - h)² (where (h, k) is the vertex)
16. Standard form of a parabola (horizontal): x - h = a(y - k)² (where (h, k) is the vertex)
17. Standard form of an ellipse (centered at the origin): (x² / a²) + (y² / b²) = 1
18. Standard form of a hyperbola (horizontal): (x² / a²) - (y² / b²) = 1
19. Standard form of a hyperbola (vertical): (y² / a²) - (x² / b²) = 1
20. Asymptotes of a hyperbola (horizontal): y = ± (b / a)x
21. Asymptotes of a hyperbola (vertical): y = ± (a / b)x
22. Distance formula in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
23. Midpoint formula in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)
24. 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)
25. Equation of a plane: Ax + By + Cz + D = 0
26. Distance from a point to a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)
22. Distance formula in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
23. Midpoint formula in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)
24. 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)
25. Equation of a plane: Ax + By + Cz + D = 0
26. Distance from a point to a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)
27. Translation: (x, y) → (x + a, y + b)
28. Reflection over the x-axis: (x, y) → (x, -y)
29. Reflection over the y-axis: (x, y) → (-x, y)
30. Reflection over the line y = x: (x, y) → (y, x)
31. Rotation by θ degrees around the origin: (x', y') = (x cosθ - y sinθ, x sinθ + y cosθ)
32. Dilation with respect to the origin: (x, y) → (kx, ky) (where k is the scale factor)
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