Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-cot Structured version   Visualization version   GIF version

Definition df-cot 46119
Description: Define the cotangent function. We define it this way for cmpt 5135, which requires the form (𝑥𝐴𝐵). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5135. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
df-cot cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-cot
StepHypRef Expression
1 ccot 46116 . 2 class cot
2 vx . . 3 setvar 𝑥
3 vy . . . . . . 7 setvar 𝑦
43cv 1542 . . . . . 6 class 𝑦
5 csin 15625 . . . . . 6 class sin
64, 5cfv 6380 . . . . 5 class (sin‘𝑦)
7 cc0 10729 . . . . 5 class 0
86, 7wne 2940 . . . 4 wff (sin‘𝑦) ≠ 0
9 cc 10727 . . . 4 class
108, 3, 9crab 3065 . . 3 class {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0}
112cv 1542 . . . . 5 class 𝑥
12 ccos 15626 . . . . 5 class cos
1311, 12cfv 6380 . . . 4 class (cos‘𝑥)
1411, 5cfv 6380 . . . 4 class (sin‘𝑥)
15 cdiv 11489 . . . 4 class /
1613, 14, 15co 7213 . . 3 class ((cos‘𝑥) / (sin‘𝑥))
172, 10, 16cmpt 5135 . 2 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
181, 17wceq 1543 1 wff cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  cotval  46122
  Copyright terms: Public domain W3C validator