โญ แ แแต 12แ แญแแ แฐแแณแ แฐแแช แจแแต แแซแแแธแ แจแแแก แจแแตแต แแญแแแแฝ!!!
1. Pythagorean theorem: aยฒ + bยฒ = cยฒ
2. Quadratic formula: x = (-b ยฑ โ(bยฒ - 4ac)) / 2a
3. Distance formula: d = โ((xโ - xโ)ยฒ + (yโ - yโ)ยฒ)
4. Slope-intercept form of a line: y = mx + b
5. Point-slope form of a line: y - yโ = m(x - xโ)
6. Midpoint formula: ((xโ + xโ)/2, (yโ + yโ)/2)
7. Law of sines: a/sin A = b/sin B = c/sin C
8. Law of cosines: cยฒ = aยฒ + bยฒ - 2ab cos C
9. Sum of angles in a triangle: A + B + C = 180ยฐ
10. Area of a triangle: A = (1/2)bh
11. Volume of a sphere: V = (4/3)ฯrยณ
12. Volume of a cylinder: V = ฯrยฒh
13. Volume of a cone: V = (1/3)ฯrยฒh
14. Surface area of a sphere: A = 4ฯrยฒ
15. Surface area of a cylinder: A = 2ฯrยฒ + 2ฯrh
16. Surface area of a cone: A = ฯrยฒ + ฯrs, where s is the slant height
17. Binomial theorem: (a + b)โฟ = ฮฃ(n choose k)a^(n-k)b^k, where ฮฃ is the sum from k=0 to n, and (n choose k) is the binomial coefficient
18. Fundamental theorem of calculus: โซa^b 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(xโฟ) = nx^(n-1)
21. Product rule for derivatives: d/dx(fg) = f'g + fg'
22. Quotient rule for derivatives: d/dx(f/g) = (f'g - fg')/gยฒ
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 in (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 in [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 in (a,b) such that f'(c) = 0
27. Integration by substitution: โซf(g(x))g'(x) dx = โซf(u) du, where u = g(x)
28. Integration by parts: โซu dv = uv - โซv du
29. L'Hopital's rule: if lim(x โ a) f(x)/g(x) = 0/0 or โ/โ, then lim(x โ a) f(x)/g(x) = lim(x โ a) f'(x)/g'(x)
30. Taylor series: f(x) = ฮฃ(n=0 to โ) f^(n)(a)/n!(x-a)^n, where f^(n) is the nth derivative of f
31. Euler's formula: e^(ix) = cos(x) + i sin(x)
32. De Moivre's theorem: (cos x + i sin x)^n = cos(nx) + i sin(nx)
33. Fundamental trigonometric identities: sinยฒ x + cosยฒ x = 1, 1 + tanยฒ x = secยฒ x, 1 + cotยฒ x = cscยฒ x
34. Double angle formulas: sin 2x = 2sin x cos x, cos 2x = cosยฒ x - sinยฒ x, tan 2x = (2tan x)/(1 - tanยฒ x)
35. Half angle formulas: sin(x/2) = ยฑโ((1 - cos x)/2), cos(x/2) = ยฑโ((1 + cos x)/2), tan(x/2) = ยฑโ((1 - cos x)/(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), sin A - sin B = 2cos((A+B)/2)sin((A-B)/2), cos A - cos B = -2sin((A+B)/2)sin((A-B)/2)
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)), sin A cos B = (1/2)(sin(A+B) + sin(A-B)), cos A sin B = (1/2)(sin(A+B) - sin(A-B))
38. Hyperbolic functions: sinh x = (e^x - e^-x)/2, cosh x = (e^x + 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 x^n = n log x
41. Exponential identities: e^x+y = e^x e^y, (e^x)^n = e^(nx), e^0 = 1
42. Binomial coefficients: (n choose k) = n!/(k!(n-k)!)
43. Pascal's triangle: each entry is the sum of the two entries above it
44. Fermat's little theorem: if p is a prime and a is not divisible by p, then a^(p-1) โก 1 (mod p)
45. Chinese remainder theorem: if mโ, mโ, ..., mโ are pairwise coprime integers and aโ, aโ, ..., aโ are any integers, then there exists an integer x that satisfies the system of congruences x โก aโ (mod mโ), x โก aโ (mod mโ), ..., x โก aโ (mod mโ)
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