lemma 7.73 $\cos 2 < 0$

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1906 · p. 230

Rests on

depends_on equation 7.53 eq:ana-cos-sin-series ¶
depends_on proposition 7.52 Alternating series test ¶
- depends_on corollary A.46 Monotone convergence ¶
  - depends_on theorem A.40 Least-upper-bound property ¶
    - depends_on definition A.36 Cut ¶
    - depends_on definition A.38 Order on $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - proves proof app:A-long-proofs@proof-29 ¶
  - proves proof app:A-long-proofs@proof-33 ¶
- depends_on definition 7.45 Series ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
- proves proof ch:05-real-analysis@proof-33 ¶
proves proof ch:05-real-analysis@proof-48 ¶

Supports

depends_on definition 7.74 $\pi$ ¶
- depends_on lemma 7.75 The first quadrant ¶
  - depends_on proposition 7.76 $\pi$ as the circle constant ¶
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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declared in the source
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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:ana-cos-sin-series	declared	parts/02-mathematical-methods/05-real-analysis.tex:1907
`depends_on`	→	Alternating series test	declared	parts/02-mathematical-methods/05-real-analysis.tex:1907
`depends_on`	←	$\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:1947
`proves`	←	ch:05-real-analysis@proof-48	declared	parts/02-mathematical-methods/05-real-analysis.tex:1910