Math 2.0

Special functions Proposition

cos pi = -1

Builds on piAngle addition formulasPythagorean trigonometric identity

Statement

cos⁡π=−1,sin⁡π=0.\cos \pi = -1, \qquad \sin \pi = 0.

Remarks

By the definition of π\pi, cos⁡(π/2)=0\cos(\pi/2) = 0. The Pythagorean identity gives sin⁡(π/2)=1\sin(\pi/2) = 1, since sin⁡>0\sin > 0 on (0,2](0, 2]. The double angle formulas then give cos⁡π=2cos⁡2(π/2)−1=−1\cos \pi = 2 \cos^2(\pi/2) - 1 = -1 and sin⁡π=2sin⁡(π/2)cos⁡(π/2)=0\sin \pi = 2 \sin(\pi/2) \cos(\pi/2) = 0. Together with sin⁡π=0\sin \pi = 0, it gives Euler's identity eiπ=−1e^{i \pi} = -1 in complex form. Mathlib states the first claim as Real.cos_pi.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

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

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Pi, Definition
About this page
Tag01A8JGR4JM
URIm2://01A8JGR4JM
Kindstatement, proposition
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 01A8JGR4JM.
Usespi, Angle addition formulas, Pythagorean trigonometric identity

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