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1. Pythagorean theorem: a² + b² = c²
2. Quadratic formula: x = (-b ± √(b² - 4ac))/2a
3. Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
4. Slope-intercept form of a line: y = mx + b
5. Point-slope form of a line: y - y₁ = m(x - x₁)
6. Midpoint formula: (((x₁ + x₂)/2, (y₁ + y₂)/2))
7. Law of sines: a/(sin A) = b/(sin B) = c/(sin C)
8. Law of cosines: c² = a² + b² - 2ab cos C
9. Sum of angles in a triangle: A + B + C = 180°
10. Area of a triangle: A = ½bh
11. Volume of a sphere: V = 4/3π r³
12. Volume of a cylinder: V = π r²h
13. Volume of a cone: V = ⅓π r²h
14. Surface area of a sphere: A = 4π r²
15. Surface area of a cylinder: A = 2π r² + 2π rh
16. Surface area of a cone: A = π r² + π rs , where s is the slant height
17. Binomial theorem: (a + b)ⁿ = ∑ₖ₌₀ⁿ n \choose k aⁿ⁻ᵏ bᵏ , where n \choose k is the binomial coefficient
18. Fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) - F(a) , where F is the antiderivative of f
19. Derivative of a constant: d/dx(c) = 0
20. Power rule for derivatives: d/dx(xⁿ) = nxⁿ⁻¹
21. Product rule for derivatives: d/dx(fg) = f'g + fg'
22. Quotient rule for derivatives: d/dx((f/g)) = (f'g - fg')/g²
23. Chain rule for derivatives: d/dx(f(g(x))) = f'(g(x))g'(x)
24. Mean value theorem: If f is continuous on [a,b] and differentiable on (a,b) , then there exists c ∈ (a,b) such that f'(c) = (f(b) - f(a))/(b-a)
25. Intermediate value theorem: If f is continuous on [a,b] , then for any y between f(a) and f(b) , there exists c ∈ [a,b] such that f(c) = y
26. Rolle's theorem: If f is continuous on [a,b] and differentiable on (a,b) , and if f(a) = f(b) , then there exists c ∈ (a,b) such that f'(c) = 0
27. Integration by substitution: ∫f(g(x))g'(x) dx = ∫f(u) du, where u = g(x)
28. Integration by parts: ∫u dv = uv - ∫v du
29. L'Hopital's rule: If lim(x → a) (f(x))/(g(x)) = 0/0 or ∞/∞, then lim(x → a) (f(x))/(g(x)) = lim(x → a) (f'(x))/(g'(x))
30. Taylor series: f(x) = ∑ₙ₌₀^∞ (f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ, where f⁽ⁿ⁾ is the nᵗʰ derivative of } f
31. Euler's formula: eⁱˣ = cos(x) + isin(x)
32. De Moivre's theorem: (cos x + isin x)ⁿ = cos(nx) + isin(nx)
33. Fundamental trigonometric identities:
• sin² x + cos² x = 1, 1 + tan² x = sec² x, 1 + cot² x = csc² x
34. Double angle formulas:
• sin 2x = 2sin x cos x, cos 2x = cos² x - sin² x, tan 2x = (2tan x)/(1 - tan² x)
35. Half angle formulas:
• sin(x/2) = ±√((1 - cos)/)2}, cos(x/2) = ±√((1 + cos)/)2}, tan(x/2) = ±√((1 - cos)/)1 + cos x}
36. Sum-to-product formulas:
• sin A + sin B = 2sin(((A+B)/2))cos(((A-B)/2)), cos A + cos B = 2cos(((A+B)/2))cos(((A-B)/2))
• Additional formulas for differences available.
37. Product-to-sum formulas:
• cos A cos B = (1/2)(cos(A-B) + cos(A+B)), sin A sin B = (1/2)(cos(A-B) - cos(A+B))
• Additional formulas for products available.
38. Hyperbolic functions:
• sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x/cosh x
39. Inverse trigonometric functions:
• arcsin x, arccos x, arctan x
40. Logarithmic identities:
• log(xy) = log x + log y, log(x/y) = log x - log y, log xⁿ = n log x
41. Exponential identities:
• eˣ⁺ʸ = eˣ eʸ, (eˣ)ⁿ = eⁿˣ, e⁰ = 1
42. Binomial coefficients:
• n choose k = (n!)/(k!(n-k)!)
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1. Pythagorean theorem: a² + b² = c²
2. Quadratic formula: x = (-b ± √(b² - 4ac))/2a
3. Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
4. Slope-intercept form of a line: y = mx + b
5. Point-slope form of a line: y - y₁ = m(x - x₁)
6. Midpoint formula: (((x₁ + x₂)/2, (y₁ + y₂)/2))
7. Law of sines: a/(sin A) = b/(sin B) = c/(sin C)
8. Law of cosines: c² = a² + b² - 2ab cos C
9. Sum of angles in a triangle: A + B + C = 180°
10. Area of a triangle: A = ½bh
11. Volume of a sphere: V = 4/3π r³
12. Volume of a cylinder: V = π r²h
13. Volume of a cone: V = ⅓π r²h
14. Surface area of a sphere: A = 4π r²
15. Surface area of a cylinder: A = 2π r² + 2π rh
16. Surface area of a cone: A = π r² + π rs , where s is the slant height
17. Binomial theorem: (a + b)ⁿ = ∑ₖ₌₀ⁿ n \choose k aⁿ⁻ᵏ bᵏ , where n \choose k is the binomial coefficient
18. Fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) - F(a) , where F is the antiderivative of f
19. Derivative of a constant: d/dx(c) = 0
20. Power rule for derivatives: d/dx(xⁿ) = nxⁿ⁻¹
21. Product rule for derivatives: d/dx(fg) = f'g + fg'
22. Quotient rule for derivatives: d/dx((f/g)) = (f'g - fg')/g²
23. Chain rule for derivatives: d/dx(f(g(x))) = f'(g(x))g'(x)
24. Mean value theorem: If f is continuous on [a,b] and differentiable on (a,b) , then there exists c ∈ (a,b) such that f'(c) = (f(b) - f(a))/(b-a)
25. Intermediate value theorem: If f is continuous on [a,b] , then for any y between f(a) and f(b) , there exists c ∈ [a,b] such that f(c) = y
26. Rolle's theorem: If f is continuous on [a,b] and differentiable on (a,b) , and if f(a) = f(b) , then there exists c ∈ (a,b) such that f'(c) = 0
27. Integration by substitution: ∫f(g(x))g'(x) dx = ∫f(u) du, where u = g(x)
28. Integration by parts: ∫u dv = uv - ∫v du
29. L'Hopital's rule: If lim(x → a) (f(x))/(g(x)) = 0/0 or ∞/∞, then lim(x → a) (f(x))/(g(x)) = lim(x → a) (f'(x))/(g'(x))
30. Taylor series: f(x) = ∑ₙ₌₀^∞ (f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ, where f⁽ⁿ⁾ is the nᵗʰ derivative of } f
31. Euler's formula: eⁱˣ = cos(x) + isin(x)
32. De Moivre's theorem: (cos x + isin x)ⁿ = cos(nx) + isin(nx)
33. Fundamental trigonometric identities:
• sin² x + cos² x = 1, 1 + tan² x = sec² x, 1 + cot² x = csc² x
34. Double angle formulas:
• sin 2x = 2sin x cos x, cos 2x = cos² x - sin² x, tan 2x = (2tan x)/(1 - tan² x)
35. Half angle formulas:
• sin(x/2) = ±√((1 - cos)/)2}, cos(x/2) = ±√((1 + cos)/)2}, tan(x/2) = ±√((1 - cos)/)1 + cos x}
36. Sum-to-product formulas:
• sin A + sin B = 2sin(((A+B)/2))cos(((A-B)/2)), cos A + cos B = 2cos(((A+B)/2))cos(((A-B)/2))
• Additional formulas for differences available.
37. Product-to-sum formulas:
• cos A cos B = (1/2)(cos(A-B) + cos(A+B)), sin A sin B = (1/2)(cos(A-B) - cos(A+B))
• Additional formulas for products available.
38. Hyperbolic functions:
• sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x/cosh x
39. Inverse trigonometric functions:
• arcsin x, arccos x, arctan x
40. Logarithmic identities:
• log(xy) = log x + log y, log(x/y) = log x - log y, log xⁿ = n log x
41. Exponential identities:
• eˣ⁺ʸ = eˣ eʸ, (eˣ)ⁿ = eⁿˣ, e⁰ = 1
42. Binomial coefficients:
• n choose k = (n!)/(k!(n-k)!)
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