Eritque Arcus Math Notes

Trigonometry

1 min read

Basic recall

  1. sin′(𝑥)=cos(𝑥)
  2. cos′(𝑥)=−sin(𝑥)
  3. sin(0)=0,cos(0)=1
  4. sin(𝜋2)=1,cos(𝜋2)=0
  5. sin(−𝑥)=−sin(𝑥)

Def 41.1

𝜋=sup{𝑥∣∀0<𝑦<𝑥,cos(𝑦)=∞}

Lemma 41.2

|sin(𝑥)|≤|𝑥|

proved by Thm 32.1 Fundamental Theorem and (∀𝑡)cos(𝑡)≤1 with

sin(𝑥)−sin(0)=∫𝑥0sin′(𝑡)𝑑𝑡=∫𝑥0cos(𝑡)𝑑𝑡≤𝑥≤|𝑥|

Then

−sin(𝑥)=sin(−𝑥)≤|𝑥|

□

Remark 41.3

1−cos(𝑥)=∫𝑥0sin(𝑡)𝑑𝑡

Graph View