Math 2.0

Special functions Theorem

Sine is 2π-periodic

Builds on Angle addition formulascos pi = -1Periodic function

Statement

For every real xx, sin⁡(x+2π)=sin⁡x,cos⁡(x+2π)=cos⁡x.\sin(x + 2\pi) = \sin x, \qquad \cos(x + 2\pi) = \cos x. The number 2π2\pi is the smallest positive period of sin⁡\sin and of cos⁡\cos.

Remarks

From cos⁡π=−1\cos \pi = -1 and sin⁡π=0\sin \pi = 0, the addition formulas give cos⁡2π=1\cos 2\pi = 1 and sin⁡2π=0\sin 2\pi = 0. Apply the addition formulas again with y=2πy = 2\pi. So sin⁡\sin and cos⁡\cos are periodic functions. Also sin⁡(x+π)=−sin⁡x\sin(x + \pi) = -\sin x and sin⁡(x+π/2)=cos⁡x\sin(x + \pi/2) = \cos x. Mathlib states the periodicity as Real.sin_periodic.

Reading path

To understand this page from the ground up, read these 65 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Periodic function definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. Bounded sequence definition
  12. Cauchy sequence definition
  13. Geometric sequence definition
  14. Limit of a sequence definition
  15. Monotone sequence definition
  16. Series definition
  17. Continuous function definition
  18. Fermat's theorem on stationary points statement
  19. Limit superior definition
  20. Cauchy sequences are bounded statement
  21. Sum of a finite geometric series statement
  22. Monotone convergence theorem for sequences statement
  23. Every sequence has a monotone subsequence statement
  24. Convergent sequence definition
  25. Pointwise convergence definition
  26. Squeeze theorem statement
  27. Cauchy product definition
  28. Partial sum definition
  29. Power series definition
  30. Intermediate value theorem statement
  31. Bolzano–Weierstrass theorem statement
  32. The sequence r^n tends to zero for |r| < 1 example
  33. Radius of convergence definition
  34. Uniform convergence definition
  35. Convergent series definition
  36. Sine and cosine definition
  37. Boundedness theorem statement
  38. Cauchy criterion for sequences statement
  39. Uniform convergence on a set definition
  40. Absolute convergence definition
  41. Series of functions definition
  42. A series of nonnegative terms converges iff its partial sums are bounded statement
  43. Sum of a series definition
  44. Extreme value theorem statement
  45. Uniformly Cauchy sequence of functions definition
  46. Geometric series example
  47. Comparison test statement
  48. Cauchy product of absolutely convergent series statement
  49. Rolle's theorem statement
  50. Root test statement
  51. Weierstrass M-test statement
  52. Angle addition formulas statement
  53. Mean value theorem statement
  54. Cauchy–Hadamard theorem statement
  55. Power series converge uniformly on smaller discs inside the radius of convergence statement
  56. A function with zero derivative is constant statement
  57. Uniform convergence of derivatives allows term-by-term differentiation statement
  58. Term-by-term differentiation statement
  59. Power series are differentiable inside the radius of convergence statement
  60. Derivative of cosine statement
  61. Derivative of sine statement
  62. Pythagorean trigonometric identity statement
  63. pi object
  64. cos pi = -1 statement
  65. Sine is 2π-periodic statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Sine_and_cosine, Graph of a function and its elementary properties
About this page
Tag0P4NQD746K
URIm2://0P4NQD746K
Kindstatement, theorem
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 0P4NQD746K.
UsesAngle addition formulas, cos pi = -1, Periodic function

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.