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Mirrors > Home > MPE Home > Th. List > df-acos | Structured version Visualization version GIF version |
Description: Define the arccosine function. See also remarks for df-asin 25603. Since we define arccos in terms of arcsin, it shares the same branch points and cuts, namely (-∞, -1) ∪ (1, +∞). (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
df-acos | ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cacos 25601 | . 2 class arccos | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 10613 | . . 3 class ℂ | |
4 | cpi 15512 | . . . . 5 class π | |
5 | c2 11771 | . . . . 5 class 2 | |
6 | cdiv 11375 | . . . . 5 class / | |
7 | 4, 5, 6 | co 7170 | . . . 4 class (π / 2) |
8 | 2 | cv 1541 | . . . . 5 class 𝑥 |
9 | casin 25600 | . . . . 5 class arcsin | |
10 | 8, 9 | cfv 6339 | . . . 4 class (arcsin‘𝑥) |
11 | cmin 10948 | . . . 4 class − | |
12 | 7, 10, 11 | co 7170 | . . 3 class ((π / 2) − (arcsin‘𝑥)) |
13 | 2, 3, 12 | cmpt 5110 | . 2 class (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) |
14 | 1, 13 | wceq 1542 | 1 wff arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) |
Colors of variables: wff setvar class |
This definition is referenced by: acosf 25612 acosval 25621 dvacos 35485 |
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